Morfismos, Vol 8, No 1, 2004

VOLUMEN 8 NÚMERO 1 ENERO A JUNIO DE 2004 ISSN: 1870-6525

Morfismos Comunicaciones Estudiantiles Departamento de Matem´aticas Cinvestav Editores Responsables • Isidoro Gitler • Jes´ us Gonz´alez

Consejo Editorial • Luis Carrera • Samuel Gitler • On´esimo Hern´andez-Lerma • Hector Jasso Fuentes • Miguel Maldonado • Ra´ ul Quiroga Barranco • Enrique Ram´ırez de Arellano • Francisco Ram´ırez Reyes • Jos´e Rosales Ortega • Mario Villalobos Arias

Editores Asociados • Ricardo Berlanga • Emilio Lluis Puebla • Isa´ıas L´opez • Guillermo Pastor • V´ıctor P´erez Abreu • Carlos Prieto • Carlos Renter´ıa • Luis Verde

Secretarias T´ecnicas • Roxana Mart´ınez • Laura Valencia

Morfismos puede ser consultada electr´onicamente en “Revista Morfismos” de la direcci´on http://www.math.cinvestav.mx. Para mayores informes dirigirse al tel´efono 50 61 38 71.

Toda correspondencia debe ir dirigida a la Sra. Laura Valencia, Departamento de Matem´aticas del Cinvestav, Apartado Postal 14-740, M´exico, D.F. 07000 o por correo electr´onico: laura@math.cinvestav.mx.

VOLUMEN 8 NÚMERO 1 ENERO A JUNIO DE 2004 ISSN: 1870-6525

Informaci´ on para Autores El Consejo Editorial de Morfismos, Comunicaciones Estudiantiles del Departamento de Matem´ aticas del CINVESTAV, convoca a estudiantes de licenciatura y posgrado a someter art´ıculos para ser publicados dentro de esta revista bajo los siguientes lineamientos • Todos los art´ıculos ser´ an enviados a especialistas para su arbitraje. No obstante, los art´ıculos ser´ an considerados s´ olo como versiones preliminares y por tanto pueden ser publicados en otras revistas especializadas.

• Se debe anexar junto con el nombre del autor, su nivel acad´emico y la instituci´ on donde estudia o labora.

• El art´ıculo debe empezar con un resumen en el cual se indique de manera breve y concisa el resultado principal que se comunicar´ a.

• Es recomendable que los art´ıculos presentados est´en escritos en Latex y sean enviados a trav´es de un medio electr´ onico. Los autores interesados pueden obtener el formato LATEX 2ε utilizado por Morfismos en “Revista Morfismos” de la direcci´ on web http://www.math.cinvestav.mx, o directamente en el Departamento de Matem´ aticas del CINVESTAV. La utilizaci´ on de dicho formato ayudar´ a en la pronta publicaci´ on del art´ıculo.

• Si el art´ıculo contiene ilustraciones o figuras, ´estas deber´ an ser presentadas de forma que se ajusten a la calidad de reproducci´ on de Morfismos.

• Los autores recibir´ an un total de 15 sobretiros por cada art´ıculo publicado.

• Los art´ıculos deben ser dirigidos a la Sra. Laura Valencia, Departamento de Matem´aticas del Cinvestav, Apartado Postal 14 - 740, M´exico, D.F. 07000, o a la direcci´ on de correo electr´ onico laura@math.cinvestav.mx

Author Information Morfismos, the student journal of the Mathematics Department of Cinvestav, invites undergraduate and graduate students to submit manuscripts to be published under the following guidelines • All manuscripts will be refereed by specialists. However, accepted papers will be considered to be “preliminary versions” in that authors may republish their papers in other journals, in the same or similar form.

• In addition to his/her affiliation, the author must state his/her academic status (student, professor,...).

• Each manuscript should begin with an abstract summarizing the main results.

• Morfismos encourages electronically submitted manuscripts prepared in Latex. Authors may retrieve the LATEX 2ε macros used for Morfismos through the web site http://www.math.cinvestav.mx, at “Revista Morfismos”, or by direct request to the Mathematics Department of Cinvestav. The use of these macros will help in the production process and also to minimize publishing costs.

• All illustrations must be of professional quality.

• 15 offprints of each article will be provided free of charge.

• Manuscripts submitted for publication should be sent to Mrs. Laura Valencia, Departamento de Matem´ aticas del Cinvestav, Apartado Postal 14 - 740, M´exico, D.F. 07000, or to the e-mail address: laura@math.cinvestav.mx

Lineamientos Editoriales “Morfismos” es la revista semestral de los estudiantes del Departamento de Matem´ aticas del CINVESTAV, que tiene entre sus principales objetivos el que los estudiantes adquieran experiencia en la escritura de resultados matem´aticos.

La publicaci´ on de trabajos no estar´ a restringida a estudiantes del CINVESTAV; deseamos fomentar tambi´en la participaci´ on de estudiantes en M´exico y en el extranjero, as´ı como la contribuci´on por invitaci´ on de investigadores.

Los reportes de investigaci´on matem´ atica o res´ umenes de tesis de licenciatura, maestr´ıa o doctorado pueden ser publicados en Morfismos. Los art´ıculos que aparecer´ an ser´ an originales, ya sea en los resultados o en los m´etodos. Para juzgar ´esto, el Consejo Editorial designar´ a revisores de reconocido prestigio y con experiencia en la comunicaci´ on clara de ideas y conceptos matem´aticos.

Aunque Morfismos es una revista con arbitraje, los trabajos se considerar´ an como versiones preliminares que luego podr´ an aparecer publicados en otras revistas especializadas.

Si tienes alguna sugerencia sobre la revista hazlo saber a los editores y con gusto estudiaremos la posibilidad de implementarla. Esperamos que esta publicaci´ on propicie, como una primera experiencia, el desarrollo de un estilo correcto de escribir matem´ aticas.

Morfismos

Editorial Guidelines “Morfismos” is the journal of the students of the Mathematics Department of CINVESTAV. One of its main objectives is for students to acquire experience in writing mathematics. Morfismos appears twice a year.

Publication of papers is not restricted to students of CINVESTAV; we want to encourage students in Mexico and abroad to submit papers. Mathematics research reports or summaries of bachelor, master and Ph.D. theses will be considered for publication, as well as invited contributed papers by researchers. Papers submitted should be original, either in the results or in the methods. The Editors will assign as referees well–established mathematicians.

Even though Morfismos is a refereed journal, the papers will be considered as preliminary versions which could later appear in other mathematical journals.

If you have any suggestions about the journal, let the Editors know and we will gladly study the possibility of implementing them. We expect this journal to foster, as a preliminary experience, the development of a correct style of writing mathematics.

Morfismos

Contenido Multiobjective Markov Control Processes: a Linear Programming Approach On´esimo Hern´ andez-Lerma and Rosario Romera . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Tutte uniqueness of locally grid graphs D. Garijo, A. M´ arquez and M.P. Revuelta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

No-inmersi´on de espacios lente Enrique Torres Giese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Morfismos, Vol. 8, No. 1, 2004, pp. 1–33

Multiobjective Markov Control Processes: a Linear Programming Approach ∗ On´esimo Hern´andez-Lerma

Rosario Romera

Abstract This paper studies discrete-time multiobjective Markov control processes (MCPs) on Borel spaces and unbounded costs. Under mild assumptions, it shows the existence of Pareto policies, which, as in multiobjective optimization problems, are also characterized as optimal policies for a certain class of single-objective (or “scalar”) MCPs. A similar result is obtained for strong Pareto policies, which are Pareto policies whose cost vector is the closest, in the Euclidean norm, to the virtual minimum. To obtain these results, the basic idea is to transform the multiobjective MCP into an equivalent multiobjective measure problem (MMP). In addition, MMP is restated as a primal multiobjective linear program and it is shown that solving the dual program is in fact the same as solving the scalarized MCPs. A multiobjective LQ example illustrates the main results.

2000 Mathematics Subject Classification: 93E20, 90C40, 90C29.

Keywords and phrases: Markov control processes, multiobjective control problems, Pareto optimality, (infinite–dimensional) multiobjective linear programming.

1

Introduction

In a standard optimal control problem there is a decision–maker or controller that wishes to optimize a single objective function. Thus, for instance, in a production control problem it is tacitly assumed that the given objective function somehow aggregates several different costs ∗ Invited Article. Research partially supported by CONACYT (M´exico) Grant 37355-E for OHL, and DGES (Spain) Grant PB96-0111 for RR.

1

2

O. Hern´andez-Lerma and R. Romera

(manufacturing costs, holding costs, distribution costs, etc.) and possibly several income sources (for example, sales, investments, and so on).

However, there are situations in which it is convenient, or perhaps even necessary, to optimize separately these functions and the controller is then led to consider a multiobjective problem of the form (say): “minimize” the cost vector V (π) := (V1 (π), . . . , Vq (π)) ∈ IRq over the class of all admissible policies π (see Section 2 for details). In particular, if π ∗ minimizes V (π) in the sense of Pareto, then π ∗ is said to be a Pareto policy. On the other hand, letting (1.1)

Vi∗ := inf Vi (π) for i = 1, . . . , q, π

and defining the virtual minimum V ∗ := (V1∗ , . . . , Vq∗ ), an important issue is to find strong Pareto policies, namely, Pareto policies π ∗ whose cost vector V (π ∗ ) is the “closest” (e.g. in the usual Euclidean norm) to V ∗ . This is the control–theoretic analogue of a goal programming problem [36] in which the goal or target is V ∗ . (We might of course consider other “goals”, but V ∗ is the most common.) Still another key problem occurs when the individual costs V1 (π), . . . , Vq (π) are ranked in order of “importance”. In this case, a lexicographically (or hierarchically) optimal policy turns out to be a particular Pareto policy.

Contributions of this paper. In this paper we study discretetime multiobjective Markov control processes (MCPs) on Borel spaces and unbounded costs. The main problems we are concerned with are the existence and characterization of both Pareto and strong Pareto policies, and also of weak and proper Pareto policies (Definition 2.5). Actually, the existence of Pareto, weak Pareto, and proper Pareto policies is very easy because it can be obtained via the usual scalarization approach, in which the multiobjective MCP is reduced to a single–objective (or standard or scalar) MCP with a “weighted” objective function of the form (1.2)

λ· V (π) := λ1 V1 (π) + · · · + λq Vq (π)

for some vectors λ in the nonnegative orthant IRq+ . However, the existence of strong Pareto policies as well as the characterization problem are more complicated, and, to the best of our knowledge, this is the

Multiobjective Markov Control Processes

3

first paper dealing with these issues for general MCPs. (See below for related literature.)

To study the latter problems we propose here to use so–called occupation measures to transform the multiobjective MCP into an equivalent multiobjective measure problem (MMP) on a suitable space of measures.

The original multiobjective control problem is thus greatly simplified because the MMP turns out to have a linear objective (vector) function defined on a convex set of measures. This implies that, for instance, the existence of strong Pareto policies essentially reduces to find the distance from the virtual minimum V ∗ to a convex set. Similarly, the characterization of Pareto policies (known as the “theorem of equivalence” in Pareto optimality [4]) can be obtained by standard convex–analytic arguments. Moreover, introducing suitable vector spaces, we reformulate the MMP as a primal multiobjective linear program and this allows us to show that solving the dual linear program is in fact the same as solving the scalar problem (1.2) using dynamic programming. As far as we can tell, this interpretation of the scalarization approach for multiobjective control problems as the dual of a multiobjective linear program has never been reported before in the literature. We should also note that to obtain the latter duality result, as well as the characterization of Pareto policies (Theorem 3.4, below) without our MMP approach would be extremely difficult — perhaps impossible — to obtain.

Related literature. Vector optimization problems can be traced back to (at least) the late 19th century; see e.g. [4, 31] for earlier references. However, according to the excellent survey by Salukvadze [35, Chapter 1], in control theory they were first introduced by Zadeh [46] in 1963. The scalarization and the hierarchical (or lexicographical) approaches were introduced by Reid and Citron [34] and Waltz [43], respectively.

Concerning multiobjective MCPs, the existence and characterization of Pareto policies have been studied by many authors but for particular classes of MCPs, for instance with a countable state space [1012,14,23,24,28,40-42,44,45], or in Borel spaces but with bounded costs [29, 37, 39]. It should be noted that for some of these classes of MCPs one can obtain very interesting results. For example, if the state and action spaces are both finite, the set of Pareto policies can be completely characterized using Theorem 1 of Arrow et al. [3], as in [14]. Moreover,

4

O. Hern´andez-Lerma and R. Romera

for finite state spaces, there are multiobjective versions of value iteration [23, 24, 45] and of policy iteration [12, 41, 42], which, as they are computationally appealing, it would be interesting to investigate if they can be extended to MCPs in uncountable spaces. On the other hand, some papers [12, 23, 24, 29] deal with a vector–minimization problem more general than ours in the sense that instead of the convex cone IR q+ , they work with the partial order induced by an arbitrary pointed convex cone in IRq . But it turns out that they restrict the control problem to some subclass of policies (e.g. deterministic stationary), whereas here we work with the set of all (randomized, history–dependent) policies. At any rate, extending our MMP approach to the case of a general pointed convex cone seems to be a purely notational problem.

Organization of the paper. The remainder of the paper is organized as follows. In Section 2 we introduce the multiobjective MCP we are concerned with, as well as the precise notions of Pareto optimality.

We consider a vector of discounted cost criteria but in Section 8 we briefly explain, among other things, how our results can be translated to average costs. In Section 3 we state our hypotheses (Assumption 3.1) and the so-called “theorem of equivalence” in Pareto optimality [4]. In fact, we state this theorem in two parts, Theorem 3.2(a) and (the converse) Theorem 3.4, because the proof of the latter requires the MMP, which is introduced until Section 4. On the other hand, Theorem 3.2(a) is the easy part of the “theorem of equivalence” and it directly yields the existence of Pareto policies. Section 3 also includes Example 3.5 on a multiobjective LQ (Linear system with Quadratic costs) MCP in which explicit Pareto policies can be calculated. In Section 5 we introduce the virtual minimum V ∗ for our multiobjective MCP, and show the existence of strong Pareto policies. We also extend a result of Tanaka [37] that can be very useful to compute strong Pareto policies; see Theorem 5.2(b). This fact is illustrated in Example 5.7, which is a continuation of the LQ Example 3.5. Section 6 presents the multiobjective Linear Programming (LP) formulation of the multiobjective MCP. The idea (as for scalar and constrained MCPs [16,17,20,21]) is to introduce suitable dual pairs of vector spaces in which the MMP (4.7) can be formulated as a multiobjective linear program. The multiobjective LP formulation is borrowed from Balb´as and Heras [7]. Section 7 contains the proof of Theorem 3.4, and, finally, in Section 8 we briefly mention some connections between our multiobjective MCPand constrained MCPs, multiobjective problems with average cost criteria, and

Multiobjective Markov Control Processes

5

multiobjective problems with “mixed” average and discounted criteria.

Remark 1.1.(Notation) If S is a Borel space (that is, a Borel subset of a complete and separable metric space), we denote its Borel σ-algebra

by B(S). If S and T are Borel spaces, then a stochastic kernel on S given T is a function (t, B) → q(B|t) from T × B(S) to the interval [0, 1] such that q(B|· ) is a measurable function on T for each fixed B ∈ B(S), and q(· |t) is a probability measure on B(S) for each fixed t.

2

Multiobjective MCPs

A multiobjective Markov control model can be represented as (2.1)

(X, A, IK, Q, (c1 , . . . , cq ), δ, γ0 ),

where X and A are Borel spaces that stand for the state space and the control (or action) set, respectively. We also have the constraint set IK, a Borel subset of X × A, and which is assumed to contain the graph of a measurable map from X to A (this ensures that the set IF in Definition 2.1, below, is nonempty). For each x ∈ X, the x-section in IK, namely A(x) := {a ∈ A|(x, a) ∈ IK}, is a (nonempty) Borel subset of A whose elements are the admissible control actions in the state x. The transition law Q is a stochastic kernel on X given IK, whereas (2.2)

c := (c1 , . . . , cq ) : IK → IRq

is a vector function whose components are used to define the different cost criteria. Finally, δ ∈ (0, 1) is a given discount factor, and γ0 is the initial distribution, a probability measure on X. If q = 1, then (2.1) will be referred to as a “scalar” (or “standard”) Markov control model.

Definition 2.1.Φ denotes the family of stochastic kernels ϕ on A given X that satisfy the constraint ϕ(A(x)|x) = 1 for all x ∈ X, and IF stands for the class of measurable functions f from X to A such that f (x) ∈ A(x) for all x ∈ X.

6

O. Hern´andez-Lerma and R. Romera

Let H0 := X, and Hn := IKn × X for n = 1, 2, . . .. A control policy is a sequence π = {πn , n = 0, 1, . . .} of stochastic kernels πn on A given Hn that satisfy the condition (2.3)

πn (A(xn )|hn ) = 1

for each “history” hn = (x0 , a0 , . . . , xn−1 , an−1 , xn ) in Hn and n = 0, 1, . . .. We denote by Π the set of control policies. A control policy π = {πn } is said to be randomized stationary if there exists ϕ ∈ Φ such that πn (· |hn ) = ϕ(· |xn ) for every history hn ∈ Hn and n = 0, 1, . . ..

The set of such policies will be identified with the family Φ in Definition 2.1. On the other hand, π = {πn } is called deterministic stationary if there exists f ∈ IF such that πn (· |hn ) is the Dirac measure concentrated at f (xn ) for all hn ∈ Hn and n = 0, 1, . . .. We shall identify IF with the collection of deterministic stationary policies.

The multiobjective MCP. Consider the control model (2.1) and let (Ω, F) be the (canonical) measurable space consisting of the sample space Ω := (X × A)∞ , and the corresponding product σ-algebra F. Then, for each policy π ∈ Π, there is a probability measure Pγπ0 and a stochastic process {(xt , at ), t = 0, 1, . . .} defined on Ω in a canonical way, where xt and at represent the state and the control variables at time t (t = 0, 1, . . .) when using the policy π. The expectation operator with respect to Pγπ0 is denoted by Eγπ0 .

For each i = 1, . . . , q and π ∈ Π, consider the δ-discounted cost (2.4)

Vi (π, γ0 ) := (1

− δ)Eγπ0

∞

δt ci (xt , at ) ,

t=0

which will be well defined under our Assumption 3.1 below. Now let V (π, γ0 ) ∈ IRq be the cost vector (2.5)

V (π, γ0 ) := (V1 (π, γ0 ), . . . , Vq (π, γ0 )).

The multiobjective control problem we are concerned with is to find a policy π ∗ that “minimizes” V (· , γ0 ) in the sense of Pareto. To state this in a precise form we first introduce some notation and terminology.

Pareto optimality. We consider IRq with the usual partial order; that is, for q-vectors u and v, the inequality u ≤ v means that ui ≤ vi

Multiobjective Markov Control Processes

7

for all i = 1, . . . , q. We also have u < v ⇔ u ≤ v and u = v; u

v ⇔ ui < vi for all i = 1, . . . , q.

A sequence {uk } ⊂ IRq converging to u is said to converge in the direction v ∈ IRq if there is a sequence of positive numbers tk such that tk → 0 and (2.6)

lim (uk − u)/tk = v.

k→∞

Let Γ be a subset of IRq . The tangent cone to Γ at u ∈ Γ, denoted T(Γ, u), is the set of all the directions v ∈ IRq in which some sequence in Γ converges to u. There are several equivalent definitions of tangent cone; see e.g. [5]. In particular, if Γ is a convex set, then ([5],p. 64) (2.7)

T(Γ, u) = closure [ t>0

1 (Γ − u) ]. t

Note that Γ − u is contained in T(Γ, u).

Definition 2.2. Let Γ be a subset of IRq . A vector u∗ in Γ is said to be (a) a Pareto point of Γ if there is no u ∈ Γ such that u < u∗ ; (b) a weak Pareto point of Γ if there is no u ∈ Γ such that u << u∗ ; (c) a proper Pareto point of Γ if u∗ is a Pareto point and, in addition, the tangent cone to Γ at u∗ does not contain vectors v < 0.

Let Par(Γ), WPar(Γ) and PPar(Γ) denote, respectively, the set of Pareto points of Γ, the set of weak Pareto points, and the set of proper Pareto points. Then (2.8)

PPar(Γ) ⊂ Par(Γ) ⊂ WPar(Γ).

Moreover, if Γ is a closed convex set, then (by Theorem 1 in [3]) Par(Γ) is contained in the closure of PPar(Γ).

8

O. Hern´andez-Lerma and R. Romera

Γ m2 u*2 u*1

m1

Figure 1.

Example 2.3.Let Γ ⊂ IR2 be as in Figure 1. Then WPar(Γ) coincides with the boundary of Γ, whereas Par(Γ) is the subset of the boundary consisting of the vector (u∗1 , m2 ) and the vectors whose first coordinate is in the half–closed interval (u∗1 , m1 ]. Finally, the proper Pareto points of Γ are the vectors in Par(Γ) with first coordinate in the open interval (u∗1 , m1 ). Also note that the vector (u∗1 , m2 ) is the lexicographical minimum of Γ in the sense of the following definition.

Definition 2.4. If u and v are vectors in IRq , u is said to be lexicographically smaller than v (in symbols: u ≤L v) if the first nonzero term of the sequence v1 − u1 , . . . , vq − uq is positive. Moreover, a vector u in Γ ⊂ IRq is called the lexicographical minimum of Γ if u ≤L u for all u ∈ Γ.

A direct application of Definitions 2.2 and 2.4 shows that the lexicographical minimum is a Pareto point.

Pareto policies. The above concepts can be extended to multiobjective MCPs in the same way as it is done for vector optimization problems [27, 31, 36]. First, as the initial distribution γ0 is fixed, we shall simplify the notation by dropping γ0 from expressions such as (2.4) and (2.5). For instance, we shall write Vi (π, γ0 ) simply as Vi (π).

Multiobjective Markov Control Processes

9

Definition 2.5.Let Γ(Π) be the set of cost vectors in (2.5), i.e.

(2.9)

Γ(Π) := {V (π)|π ∈ Π}.

A policy π ∗ ∈ Π is said to be

(a) a Pareto policy (respectively, a weak Pareto policy or a proper Pareto policy) if its corresponding cost vector V (π ∗ ) is in Par(Γ(Π)) (respectively, in WPar(Γ(Π)) or in PPar(Γ(Π)));

(b) lexicographically optimal if V (π ∗ ) is the lexicographical minimum of Γ(Π).

In other words, π ∗ ∈ Π is a Pareto policy (or Pareto optimal) if there is no policy π such that V (π) < V (π ∗ ), and similarly for weak or proper Pareto policies.

The set Γ(Π) in (2.9) is called the performance set (also known as the objective or achievable set) of the multiobjective MCP. An example in which Γ(Π) is similar to the set Γ in Figure 1 is given in [18], where it is shown that the so–called cµ–rule for priority queues is lexicographically optimal — hence a nonproper Pareto policy. In fact, there are many examples of lexicographically optimal policies, including Blackwell optimal policies [26], bias optimal policies [21, 25], and average cost optimal policies that in addition minimize the cost variance [22].

Remark 2.6.(a) To find lexicographically optimal policies we may proceed as follows. Let Π0 := Π, and for i = 1, . . . , q let (2.10)

Vi := inf{Vi (π)|π ∈ Πi−1 },

and, finally, let Πi be the set of policies in Πi−1 that attain the minimum in (2.10). Then, assuming that the sets Πi are nonempty, Πq consists of the lexicographically optimal policies. Moreover, if π is in Πq , then its cost vector V (π) = (V1 , . . . , Vq ) is of course the lexicographical minimum of Γ(Π). (b) The procedure in (2.10) is also valid for q = ∞, that is, for infinite cost vectors V (π), as in Blackwell optimality [26], for instance. (c) If for some i = 1, . . . , q the set Πi in (a) consists of a single policy πi , then πi is the unique lexicographically optimal policy. (d) As in (2.8), PPar(Γ(Π)) ⊂ Par(Γ(Π)) ⊂ WPar(Γ(Π)).

10

3

O. Hern´andez-Lerma and R. Romera

Pareto optimal policies

To study the existence and characterization of Pareto policies, in the remainder of the paper we impose the following assumption.

Assumption 3.1.The multiobjective Markov control model (2.1) satisfies that: (a) The constraint set IK ⊂ X × A is closed. (b) The functions ci are nonnegative and lower semicontinuous and, moreover, at least one of them, say c1 , is inf-compact, which means that for each r ∈ IR, the level set Kr := {(x, a) ∈ IK|c1 (x, a) ≤ r}

(3.1) is compact.

(c) The transition law Q is weakly continuous; that is, denoting by Cb (S) the space of continuous bounded functions on a topological space S, the map (3.2) (x, a) →

X

h(y)Q(dy|x, a) is in Cb (IK) for each h ∈ Cb (X).

(d) There exists a policy π ∈ Π such that Vi (π) < ∞ for all i = 1, . . . , q. (Recall that Vi (π, γ0 ) ≡ Vi (π).)

Observe that Assumption 3.1 is not restrictive at all. In fact, it holds in most applications to queueing systems, productions models, etc. In particular, Assumption 3.1(c) holds if the state process {xt } evolves according to a discrete-time equation of the form xt+1 = G(xt , at , ξt ), t = 0, 1, . . . , where the ξt are i.i.d. disturbances independent of the initial state x0 , and G(x, a, s) is a given measurable function, continuous in (x, a) ∈ IK for each s. This class of systems includes the LQ problem in Examples 3.5 and 5.7, below.

The existence problem. To study the existence of Pareto policies we shall first follow the well-known “scalarization” approach. Thus,

Multiobjective Markov Control Processes

11

given a q–vector λ > 0 we consider the scalar (or real-valued) cost-perstage function q

cλ (x, a) := λ· c(x, a) =

(3.3)

λi ci (x, a), i=1

and, as in (2.4), we consider a δ-discounted cost V λ (π) ≡ V λ (π, γ0 ) with λ

V (π) := (1 −

(3.4)

∞

δ)Eγπ0

δt cλ (xt , at ) .

t=0

Using (3.3) and (2.5) we may write V λ (π) as q

V λ (π) = λ· V (π) =

(3.5)

λi Vi (π). i=1

It is clear that minimizing V λ (· ) over Π is equivalent to minimize V λ (· ) multiplied by a positive constant. Hence, occasionally we shall assume that the vector λ in (3.3)-(3.5) belongs to the set q

Λ := {λ ∈ IRq++ |

(3.6)

λi = 1}, i=1

IRq++

is the set of vectors λ 0. We may then state an existence where result as follows. (Observe that part (d) in Theorem 3.2 gives a little more than the existence of Pareto policies because, in fact, it ensures the existence of deterministic stationary Pareto policies.)

Theorem 3.2. Suppose that for some q–vector λ = (λ1 , . . . , λq ) > 0 there is a policy π ∗ ∈ Π that is optimal for the scalar criterion (3.4), i.e.

V λ (π ∗ ) ≤ V λ (π)

(3.7)

∀ π ∈ Π.

Then: (a) π ∗ is a weak Pareto policy. (b) If in addition (3.8)

V λ (π ∗ ) < V λ (π)

then π ∗ is a Pareto policy.

∀π∈Π

with

V (π) = V (π ∗ ),

12

O. Hern´andez-Lerma and R. Romera

(c) If λ 0 (in particular if λ is in Λ), then π ∗ is a proper Pareto policy. (d) If λ1 > 0, then there exists a deterministic stationary policy fλ ∈ IF that is a weak Pareto policy; if, moreover, π ∗ ≡ fλ satisfies (3.8), then fλ is a Pareto policy. Finally, if λ 0, then fλ is a proper Pareto policy.

Proof: (a) Suppose that π ∗ is not a weak Pareto policy. Then there exists a policy π ∈ Π such that V (π) V (π ∗ ) and, therefore, as λ > 0, λ λ ∗ we get V (π) < V (π ), which contradicts (3.7). (b) Similarly, if π ∗ is not a Pareto policy, there exists π ∈ Π such that V (π) < V (π ∗ ). Hence V λ (π) ≤ V λ (π ∗ ), which contradicts (3.8). (c) If π ∗ is not a proper Pareto policy, then the tangent cone to Γ(Π) at V (π ∗ ), i.e. T (Γ(Π), V (π ∗ )), contains a vector v < 0. Therefore, there exists a sequence {π k } in Π and a sequence {tk } of positive numbers such that lim (V (π k ) − V (π ∗ ))/tk = v. k→∞

As λ

0, we have λ · v < 0. It follows that for all k sufficiently large λ · (V (π k ) − V (π ∗ )) = V λ (π k ) − V λ (π ∗ ) < 0,

which contradicts (3.7). (d) Suppose that λ1 > 0. Then, by Assumption 3.1(b), λ1 · c1 (x, a) is nonnegative and inf–compact, and, therefore (by (3.3) and the first part of Assumption 3.1(b)), so is cλ . The latter fact together with Assumption 3.1(a),(c),(d) implies the existence of a deterministic stationary policy π ∗ ≡ f λ that satisfies (3.7); see e.g. [15] or Theorem 4.2.3 in [20]. Hence, by part (a), f λ is a weak Pareto policy. The remaining statements in (d) are proved similarly.

To obtain the converse of parts (a),(b),(c) in Theorem 3.2 we will use a special reformulation (introduced in Section 4) of the original multiobjective MCP. This requires to restrict the “admissible” policies to the following set.

Definition 3.3.Π0 denotes the set of policies π ∈ Π for which Vi (π) < ∞ for all i = 1, . . . , q.

Multiobjective Markov Control Processes

13

By our Assumption 3.1(d), the set Π0 is nonempty. The following theorem is proved in Section 7.

Theorem 3.4. Let π ∗ be a policy in Π0 . If π ∗ is a weak Pareto policy, then there exists a q–vector λ > 0 for which (3.7) holds. If π ∗ is a proper Pareto policy, then λ 0.

As a Pareto policy is weak Pareto (recall (2.8)), Theorem 3.4 tacitly includes the case in which π ∗ is a nonproper Pareto policy. Thus Theorem 3.4 is a (slight) extension of the so–called “theorem of equivalence” in Pareto optimality [4]. Finally, observe that Theorems 3.4 and 3.2 indeed characterize weak and proper Pareto policies because they yield that, for instance, π ∗ ∈ Π0 is a proper Pareto policy if and only if π ∗ minimizes the scalar criterion (3.4) for some q–vector λ 0.

The following example illustrates Theorem 3.2.

Example 3.5. Let α and β be nonzero real numbers and consider the scalar linear system (3.9)

xt+1 = αxt + βat + ξt for t = 0, 1, . . . ,

with state and control spaces X = A = IR. The disturbances ξt are i.i.d. random variables, independent of the initial state x0 , and such that (3.10)

E(ξ0 ) = 0 and E(ξ02 ) =: σ 2 < ∞.

For i = 1, . . . , q, let si and ri be strictly positive numbers, and let ci (x, a) be the quadratic cost (3.11)

ci (x, a) := si x2 + ri a2 .

Then, for each q–vector λ > 0, the scalar problem (3.3)-(3.5) corresponds to the linear system (3.9) with quadratic cost (3.12)

cλ (x, a) = (λ· s)x2 + (λ· r)a2

with s := (s1 , . . . , sq ) and r := (r1 , . . . , rq ). Moreover, for each i = 1, . . . , q, let zi be the unique positive solution of the Riccati equation (3.13)

δβ 2 z 2 + (ri − ri α2 δ − si β 2 δ)z − si ri = 0.

14

O. Hern´andez-Lerma and R. Romera

Now replace si and ri with the coefficients λ· s and λ· r in (3.12), respectively, and let z(λ) be the corresponding unique positive solution of (3.13). Then, as is well-known (see, for instance, p. 72 in [20]), the optimal control policy fλ ∈ IF for the scalar problem is (3.14)

fλ (x) = − λ· r + δβ 2 z(λ)

−1

αβδz(λ)x ∀x ∈ X,

and, moreover, for each initial state x0 = x, the optimal cost function is (3.15)

V λ (fλ , x) = z(λ) (1 − δ)x2 + δσ 2

∀x ∈ X,

with σ 2 as in (3.10). Therefore, assuming that the initial distribution γ0 satisfies that (3.16)

γ 0 :=

x2 γ0 (dx) < ∞,

the optimal cost V λ (fλ ) ≡ V λ (fλ , γ0 ) in the left-hand side of (3.7) is obtained by integrating both sides of (3.15) with respect to γ0 . This yields (3.17)

V λ (fλ ) = k(γ0 )z(λ),

with

k(γ0 ) := (1 − δ)γ 0 + δσ 2 .

0, and a weak

By Theorem 3.2, fλ is a proper Pareto policy if λ Pareto policy if λ > 0. In particular, let e(i) be the unit vector with coordinates ei (i) = 1 and ej (i) = 0 for j = i. Then replacing λ in (3.17) with e(i) we obtain the “partial” minimum cost in (1.1), i.e.

(3.18)

Vi∗ := inf Vi (π) = Vi (fe(i) ) = k(γ0 )zi π

∀ i = 1, . . . , q.

This gives the virtual minimum V ∗ = (V1∗ , . . . , Vq∗ ), which is illustrated in Figure 2 for the case q = 2. In that figure, the Pareto set Par(Γ(Π)) is the part of the boundary of Γ(Π) with first coordinate in [V1∗ , V1 (fe(2) )].

On the other hand, by the uniqueness of optimal policies for LQ (linear– quadratic) systems, it follows from Remark 2.6(a),(c) that fe(1) is the lexicographically optimal policy, whose corresponding cost vector V := V (fe(1) ) has coordinates Vi = Vi (fe(1) ) for all i = 1, . . . , q, i.e.

(3.19)

V = (V1∗ , V2 (fe(1) ), . . . , Vq (fe(1) )).

Multiobjective Markov Control Processes

15

V2 (π)

V2 (fe (1 ))

Γ(Π)

V2 * V1 *

V1 (fe ( 2 ) )

V1 (π)

Figure 2. See (3.18), (3.19).

Remark 3.6.Consider a single, or scalar, LQ system with cost c(x, a) = sx2 + ra2 ; see (3.11). If the coefficients s and r are both positive, then an optimal policy for this problem can be interpreted as a proper Pareto policy for a two–dimensional multiobjective control problem with individual costs c1 (x, a) := x2 and c2 (x, a) := a2 . In fact, a similar interpretation is valid for any scalar control problem with additive costs, say of the form c(x, a) = r1 c1 (x, a) + · · · + rq cq (x, a) with positive coefficients r1 , . . . , rq . See [19] for details.

4

A multiobjective measure problem

In this section we reformulate the multiobjective MCP as an equivalent multiobjective measure problem (MMP) on a suitable vector space of measures. This reformulation greatly simplifies the proofs of some results and, in addition, it can be used to write the multiobjective MCP as a multiobjective linear program (see Section 6).

Occupation measures. For each policy π ∈ Π, let µπ ≡ µπγ0 be the corresponding δ-discount expected occupation measure, which is defined

16

O. Hern´andez-Lerma and R. Romera

as (4.1)

µπ (D) := (1 − δ)

∞ t=0

δt Pγπ0 (xt , at ) ∈ D

∀D ∈ B(X × A).

This is a probability measure on X × A that, by (2.3), is concentrated on IK. Moreover, if π is in Π0 (see Definition 3.3), then a standard argument (see, for instance, Remark 9.4.2(b) in [21, p. 85]) yields that Vi (π) in (2.4) can be written as (4.2)

Vi (π) = µπ , ci :=

ci dµπ (i = 1, . . . , q).

IK

To state other properties of occupation measures we shall use the following notation: if µ is a finite signed measure on X × A, we denote its variation by |µ| = µ+ + µ− , and its marginal (or projection) on X by µ, that is, µ(B) := µ(B × A) ∀B ∈ B(X).

We also introduce the following sets of measures.

Definition 4.1.M (IK) denotes the vector space of finite signed measures on X × A, concentrated on IK, and such that (4.3)

|µ|, ci =

ci d|µ| < ∞ ∀i = 1, . . . , q.

Further, M+ (IK) ⊂ M (IK) stands for the convex cone of nonnegative measures in M (IK), and Mδ (IK) ⊂ M+ (IK) is the subfamily of nonnegative measures for which (4.4)

µ(B) = (1 − δ)γ0 (B) + δ

IK

Q(B|x, a)µ(d(x, a)) ∀B ∈ B(X).

As µ(X) = µ(X × A), it is evident from (4.4) that (4.5)

Mδ (IK) is a convex set of probability measures.

It also turns out that Mδ (IK) coincides with the family of occupation measures in (4.1). More precisely (as in [15, pp. 386-387] or [20, Theorem 6.3.7], for instance), we have the following result in which Π0 is as in Definition 3.3.

Lemma 4.2. If π is a policy in Π0 , then its occupation measure µπ is in Mδ (IK). Conversely, if µ is in Mδ (IK), then µ is the occupation measure of a policy in Π0 (that is, there exists π ∈ Π0 such that µπ = µ).

Multiobjective Markov Control Processes

17

For µ ∈ Mδ (IK) and c as in (2.2), let (4.6)

µ, c := ( µ, c1 , . . . , µ, cq ).

Now consider the following multiobjective measure problem (MMP): (4.7)

minimize { µ, c |µ ∈ Mδ (IK)}.

By (4.2) and Lemma 4.2, MMP is equivalent to our original multiobjective MCP if we restrict ourselves — which we do in the rest of this paper — to the set (4.8)

Γ(Π0 ) := {V (π)|π ∈ Π0 }

in lieu of the set Γ(Π) in (2.9). On the other hand, from (4.2), (4.5) and Lemma 4.2 we may immediately conclude the following.

Lemma 4.3. Γ(Π0 ) can be expressed as (4.9)

Γ(Π0 ) = { µ, c |µ ∈ Mδ (IK)},

which is a convex subset of IRq+ .

Actually, the convexity of Γ(Π0 ) is a well–known fact (see e.g. [10, 33, 38]). However, we wish to emphasize here that this convexity is a straightforward, trivial, consequence of the MMP formulation: see (4.6) and (4.5). This illustrates the advantage of using the MMP instead of the original multiobjective MCP. In the following section we use the MMP (4.7) to show the existence of “strong” Pareto policies, and in Section 7 we use it to prove Theorem 3.4.

5

Strong Pareto optimality

For each i = 1, . . . , q, let Vi∗ ≡ Vi∗ (γ0 ) be the optimal δ-discounted cost of the scalar MCP with cost-per-stage ci (x, a), that is, Vi∗ := inf Vi (π) (with Vi (π) as in (2.4)). π

The q-vector V ∗ := (V1∗ , . . . , Vq∗ ) is called the virtual minimum for the multiobjective MCP. (V ∗ is also known as the utopian or the ideal or

18

O. Hern´andez-Lerma and R. Romera

the shadow minimum.) Let · be the Euclidean norm in IRq , and let ρ : Π0 → IR+ be the map defined as ρ(π) := V (π) − V ∗

(5.1)

for π ∈ Π0 .

This is a utility function (or a strongly monotonically increasing function [27]) for the multiobjective MCP in the sense that if π and π are such that V (π) < V (π ), then ρ(π) < ρ(π ). (In (5.1) we took the Euclidean norm to fix ideas, but in fact we may take any norm in IRq . See Remark 5.6.)

Definition 5.1.A policy π ∗ ∈ Π0 is said to be strong Pareto optimal (or a strong Pareto policy) if it minimizes the function ρ, that is, ρ(π ∗ ) = inf{ρ(π)|π ∈ Π0 } =: ρ∗ .

(5.2)

As ρ is a utility function, it is clear that a strong Pareto policy is Pareto optimal, but of course the converse is not true.

Let Γ(Π0 ) be as in (4.8). For each λ ∈ IRq , let ∆(λ) := inf{λ· (V (π) − V ∗ )|π ∈ Π0 }

(5.3)

be the so-called support function of Γ(Π0 ) − V ∗ at λ. Moreover, let S ⊂ IRq be the closed unit sphere centered at the origin, and let S1 be its boundary, i.e., S := {λ | λ

1} and S1 := {λ | λ = 1}

Theorem 5.2. Suppose that ρ∗ > 0. Then: (a) There exists a strong Pareto policy; (b) There exists a vector λ∗ ∈ S1 ∩ IRq++ such that (5.4)

ρ∗ = ∆(λ∗ ) = max ∆(λ) λ∈S

and, moreover, for any strong Pareto policy π ∗ , the vector λ∗ is “aligned” with V (π ∗ ) − V ∗ , i.e.

(5.5)

λ∗ · (V (π ∗ ) − V ∗ ) = λ∗

V (π ∗ ) − V ∗ = ρ∗ .

Multiobjective Markov Control Processes

19

For completeness and ease of reference, before proving Theorem 5.2 we state some well-known technical facts. The following lemma can be obtained from the definition of inf–compactness and Prohorov’s Theorem [8].

Lemma 5.3. Let Y be a metric space and M a family of probability measures on Y . If there exists a nonnegative and inf-compact function v on Y such that sup{ µ, v |µ ∈ M } < ∞, then M is relatively compact, that is, for each sequence {µn } in M there is a probability measure µ on Y and a subsequence {µm } of {µn } such that µm converges weakly to µ in the sense that (5.6)

µm , u

µ, u

u ∈ Cb (Y ).

Lemma 5.4. Let Y be a metric space, and v : Y → IR lower semicontinuous and bounded below. If µm and µ are probability measures on Y and µm converges weakly to µ (that is, as in (5.6)), then lim inf µm , v

(5.7)

m→∞

µ, v .

Lemma 5.4 is well known: see, for instance, statement (12.3.37) in [21, p. 225].

Lemma 5.5. The set Mδ (IK) (in Definition 4.1) is closed with respect to the topology of weak convergence.

Proof: Let {µm } be a sequence in Mδ (IK) such that µm converges weakly to µ. Choose an arbitrary function h in Cb (X). By (3.2), h(y)Q(dy|· ) is in Cb (IK), and, therefore, by the weak convergence of µm to µ, we get h(y)Q(dy|x, a)µm (d(x, a)) →

h(y)Q(dy|x, a)µ(d(x, a)).

Similarly, the marginals µm converge weakly to the marginal µ. Hence, as each µm satisfies (4.4), so does the limiting probability measure µ.

Thus, to complete the proof that µ is in Mδ (IK), it only remains to show that (4.3) holds for µ. This, however, follows from Assumption 3.1(b) and Lemma 5.4, which together yield lim inf µm , ci m→∞

µ, ci

i = 1, . . . , q.

20

O. Hern´andez-Lerma and R. Romera

This implies that µ satisfies (4.3).

Proof of Theorem 5.2. (a) By (4.6) and Lemma 4.2, we may express ρ∗ in (5.2) as ρ∗ = inf{

µ, c

V∗

|µ ∈ Mδ (IK)}.

Now let {µn } be a sequence in Mδ (IK) such that, as n → ∞, (5.8)

µn , c

V∗

ρ∗ .

Choose an arbitrary ε > 0 and let n(ε) be such that µn , c

V∗

ρ∗ + ε ∀ n ≥ n(ε).

This implies the existence of a constant k such that µn , ci n ≥ n( ) and i = 1, . . . , q. In particular, (5.9)

µn , c1

k for all

k ∀n ≥ n( ).

Thus, as c1 is inf-compact (Assumption 3.1(b)), (5.9) and Lemma 5.3 imply the existence of a subsequence {µm } of {µn } and a probability measure µ∗ on X × A, concentrated on IK (by Assumption 3.1(a)), such that µm converges weakly to µ∗ . By Lemma 5.5, µ∗ is in Mδ (IK), and, by (5.7) and (5.8), (5.10)

µ∗ , c

V ∗ = ρ∗ .

Finally, let π ∗ ∈ Π0 be the policy associated to µ∗ , and use (4.2) to rewrite (5.10) as V (π ∗ ) − V ∗ = ρ∗ . This completes the proof of part (a). (b) If π ∗ ∈ Π0 is strong Pareto optimal, then the support function in (5.3) becomes ∆(λ) = λ· (V (π ∗ ) − V ∗ ),

and the vector λ∗ := (V (π ∗ ) − V ∗ )/ V (π ∗ ) − V ∗ satisfies (5.4) and (5.5).

Remark 5.6.By the convexity of Γ(Π0 ) (Lemma 4.3), finding a strong Pareto policy essentially reduces to the problem of finding the distance from the virtual minimum V ∗ to the convex set Γ(Π0 ). This yields, in particular, that part (b) in Theorem 5.2 can be seen as a special case

Multiobjective Markov Control Processes

21

of the “Minimum Norm Duality” in Luenberger [32, p. 136, Theorem 1]. Hence, as the latter result is true for an arbitrary normed linear space, in (5.1) we may take any norm instead of the Euclidean one. For instance, one could take a weighted p –norm, with 1 ≤ p ≤ ∞, which is very common in vector optimization [27, 31, 36].

Example 5.7. (Example 3.5 continued). Consider again the LQ problem (3.9)–(3.11). For each i = 1, . . . , q, let Vi∗ = k(γ0 )zi be the partial minimum in (3.18), where zi is the unique positive solution of (3.13).

Thus, letting z ∗ := (z1 , . . . , zq ), the LQ problem’s virtual minimum V ∗ = (V1∗ , . . . , Vq∗ ) becomes V ∗ = k(γ0 )z ∗ .

(5.11)

Moreover, to find a strong Pareto policy we may proceed as follows.

From (5.11) and (3.17), the support function in (5.3) is given by ∆(λ) = k(γ0 )[z(λ) − λ· z ∗ ] ∀λ ∈ IRq .

Now let λ∗ ∈ S1 ∩ IRq++ be as in Theorem 5.2(b). Then a strong Pareto policy is obtained from (3.14) taking λ = λ∗ , and the cost vector “closest” to V ∗ is given by (3.17) with λ = λ∗ .

6

The multiobjective LP approach

In this section we follow Balb´as and Heras [7] to formulate our multiobjective MCP as a multiobjective linear program. This requires to introduce two dual pairs (M (IK), F (IK)) and (M (X), F (X)) of vector spaces, which are essentially the same as those defined in [20, §6.3] or [21, §12.3]. (The reader may consult the latter references or [2] for general facts on infinite-dimensional scalar linear programming (LP).)

Define w : IK → IR++ as (6.1)

w(x, a) := 1 + c1 (x, a) + · · · + cq (x, a).

(More generally, our approach may use any nonnegative “weight” function w(x, a) provided that it is bounded away from zero and that it majorizes all of the functions ci (x, a). Thus, instead of w in (6.1) we could use, for instance, w := + max(c1 , . . . , cq ) for any > 0.) Observe that (4.3) is equivalent to (6.2)

w d|µ| < ∞.

22

O. Hern´andez-Lerma and R. Romera

Therefore, the vector space M (IK) can be described as the space of finite signed measures µ on X × A, concentrated on IK, and for which (6.2) holds.

Now let F (IK) be the vector space of real-valued measurable functions v on IK such that v

(6.3)

w

:= sup |v(x, a)|/w(x, a) < ∞. (x,a)

From (6.1) it follows that each of the cost functions ci belongs to F (IK), and, on the other hand, (M (IK), F (IK)) is a dual pair of vector spaces with respect to the bilinear form (6.4)

µ, v :=

v dµ for µ ∈ M (IK), v ∈ F (IK).

We also consider another dual pair (M (X), F (X)) defined exactly as above but replacing IK and w with X and w0 (x) := inf w(x, a) ∀x ∈ X, a∈A(x)

respectively.

Weak topologies. Henceforth we consider M (IK) to be endowed with the weak toplogy σ(M (IK), F (IK)), which will be referred to as the σ-weak topology. Thus a sequence (or a net) {µn } σ-converges to µ if (6.5)

µn , v

µ, v

v ∈ F (IK).

This should not be confused with the “weak convergence” (5.6), which is restricted to continuous and bounded functions. (Note that, of course, Cb (IK) ⊂ F (IK).) The vector spaces F (IK), M (IK), and F (X) are also endowed with the corresponding σ-weak topologies.

In the remainder of this section we suppose that Assumption 3.1 and the following Assumption 6.1 are both satisfied.

Assumption 6.1. stant k, X

X w0 (y)Q(dy|· )

is in F (IK); that is, for some con-

w0 (y)Q(dy|x, a) ≤ kw(x, a) ∀(x, a) ∈ IK.

Multiobjective Markov Control Processes

23

Assumptions 6.1 and 3.1(d) ensure, in particular, that the initial distribution γ0 is in the space M (X).

Let L : M (IK) → M (X) be the linear map µ → Lµ defined as (6.6)

(Lµ)(B) := µ(B) − δ

Q(B|x, a)µ(d(x, a)).

IK

The adjoint L∗ : F (X) → F (IK) of L, that is, the linear map L∗ for which (6.7)

Lµ, u = µ, L∗ u

µ ∈ M (IK), u ∈ F (X),

is given by (6.8)

(L∗ u)(x, a) = u(x) − δ

X

u(y)Q(dy|x, a) ∀(x, a) ∈ IK.

By Assumption 6.1, L∗ indeed maps F (X) into F (IK), which is equivalent to say that L is σ–weakly continuous.

Multiobjective LP. For each µ in M (IK), let µ, c be as in (4.6) and consider the primal program (PP): minimize µ, c (6.9)

subject to: Lµ = (1 − δ)γ0 , µ ∈ M+ (IK).

Comparing (PP) with the MMP (4.7) we can see that they are essentially the same but the former has a little more “structure”: the constraint (4.4) has been rewritten in (6.9) using the σ-weakly continuous map L. A feasible solution µ∗ for (PP) is said to be optimal if there is no feasible µ such that µ, c < µ∗ , c . If such an optimal solution exists, then (PP) is said to be solvable. Thus, from Theorem 3.2(d) and the equivalence of (4.7) and the multiobjective MCP, we conclude the following.

Corollary 6.2. (PP) is solvable.

To state the dual program we need some notation. Let F (X)q be the vector space of IRq -valued functions u = (u1 , . . . , uq ) with ui ∈ F (X)

24

O. Hern´andez-Lerma and R. Romera

for all i = 1, . . . , q. For u ∈ F (X)q and λ ∈ IRq , let uλ ∈ F (X) and L∗ u ∈ F (IK)q be the functions given by q

(6.10)

λi ui , and L∗ u := (L∗ u1 , . . . , L∗ uq ),

λ

u := λ· u = i=1

respectively. Moreover, if ν is in M (X), we write ν, u := ( ν, u1 , . . . , ν, uq ).

Then, from [7, p. 380], we can see that the dual program (DP) of (PP) is as follows: (DP) maximize (1 − δ)γ0 , u

(6.11) subject to: λ· L∗ u ≤ λ· c with u ∈ F (X)q , for some λ ∈ IRq++ .

In fact, if we let Fλ := {u ∈ F (X)q |λ· Lµ, u

λ· µ, c

µ ∈ M+ (X)}

and use (6.7), it then follows that the dual constraint (6.11) can be expressed as in [7], namely: u is in Fλ for some λ ∈ IRq++ .

On the other hand, using (6.10) and (6.8) we can write (6.11) in the more explicit form (6.12)

uλ (x) ≤ cλ (x, a) + δ

for some λ ∈ (6.13)

IRq++ .

X

uλ (y)Q(dy|x, a) ∀(x, a) ∈ IK,

The latter inequality yields

uλ (x) ≤ min

a∈A(x)

cλ (x, a) + δ X

uλ (y)Q(dy|x, a) ∀x ∈ X,

which, when the equality holds, that is, (6.14)

uλ (x) = min

a∈A(x)

cλ (x, a) + δ X

uλ (y)Q(dy|x, a) ∀x ∈ X.

becomes the dynamic programming equation (d.p.e.) for the scalar MCP with cost function (1 − δ)−1 V λ (π, x), where V λ (π, x) is the function in (3.5) when the initial state is x0 = x.

Multiobjective Markov Control Processes

Remark 6.3.Let V∗λ (x) := inf π V λ (π, x) for all x ∈ X. Then (1 − δ)−1 V∗λ (x) is the (pointwise) minimal solution of the d.p.e. (6.14).

Moreover, if V∗λ is in F (X) and uλ satisfies (6.12)-(6.13), then wellknown arguments (see [20, Lemma 4.2.7], for instance) give that (6.15)

uλ (x) ≤ (1 − δ)−1 V∗λ (x) ∀x ∈ X,

and for this reason uλ is said to be a subsolution of the d.p.e. (6.14).

Note that (6.15) yields (6.16)

(1 − δ)γ0 , uλ

γ0 , V∗λ .

Therefore (by the equivalence of (6.11) and (6.12)), we can see the dual program (DP) as the problem of maximizing integrals as in the lefthand side of (6.16) over the family of subsolutions uλ of the d.p.e. for a class of scalar MCPs parameterized by λ ∈ IRq++ . Thus, the multiobjective LP formulation gives us a “primal-dual” interpretation of the relation between our original multiobjective MCP and the scalar MCPs in (3.3)-(3.5). This interpretation can also be obtained from the “complementary slackness” property in the following proposition from [7] adapted to our current situation.

Proposition 6.4. Let µ be a feasible solution for (PP) and u a feasible solution for (DP). Then (a) (Weak duality.) We never have (1 − δ)γ0 , u > µ, c . (b) (Complementary slackness.) If in addition (6.17)

µ, c − L∗ u = 0,

then µ is optimal for (PP) and u is optimal for (DP).

Proof: Part (a) is straightforward, and in turn (a) implies (b) because, by (6.7) and (6.9), we can write (6.17) as (1 − δ)γ0 , u = µ, c .

Now, to obtain the primal-dual interpretation mentioned in the last part of Remark 6.3, it suffices to note that (6.17) is equivalent to (6.18)

µ, cλ − L∗ uλ = 0 ∀λ ∈ IRq++ .

25

26

O. Hern´andez-Lerma and R. Romera

In fact, by (6.8), we can recognize the integrand cλ − L∗ uλ in (6.18) as the difference between the two sides of (6.12). Therefore, we can obtain a solution (µ, uλ ) for (6.18) in the obvious manner: choose an arbitrary λ ∈ IRq++ and let V∗λ be as in Remark 6.3. Let uλ∗ (x) := (1 − δ)−1 V∗λ (x) ∀x ∈ X. Furthermore (as in the proof of Theorem 3.2(d)), let f∗ ∈ IF be a stationary policy such that f∗ (x) ∈ A(x) attains the minimum in the d.p.e.

(6.14) for all x ∈ X, and, finally, let µ∗ be the occupation measure associated with f∗ . Then, by their very definitions, it follows that µ∗ is feasible for (PP), uλ∗ is feasible for (DP), and µ∗ , cλ − L∗ uλ∗ = 0.

7

Proof of Theorem 3.4

Let us first suppose that π ∗ ∈ Π0 is a proper Pareto policy. Let µ∗ ∈ Mδ (IK) be the occupation measure corresponding to π ∗ (see (4.1)). By (4.2), (4.6) and Lemma 4.2, to prove Theorem 3.4 it suffices to show the 0 such that existence of a q–vector λ λ · µ∗ , c

λ · µ, c

µ ∈ Mδ (IK)

(cf. (3.7)) or, equivalently, (7.1)

µ − µ∗ , cλ

0 ∀ µ ∈ Mδ (IK),

with cλ as in (3.3). With this in mind, consider the set Γ(Π0 ) in (4.9), and let T0 := T (Γ(Π0 ), µ∗ , c ) be the tangent cone to Γ(Π0 ) at µ∗ , c .

As Γ(Π0 ) is convex (Lemma 4.3), we have (7.2)

µ∗ , c

Γ(Π0 )

T0 .

(Recall (2.7).) Let B be the set of q–vectors u < 0 such that q

(7.3) i=1

ui = −1,

and note that T0 − B is a convex set that does not contain the vector zero. Therefore, by a well–known separation theorem (e.g. [5, p. 30], [32, p. 133], there exists a vector λ = 0 such that λ · (v − u) > 0

∀ v ∈ T0 , u ∈ B.

Multiobjective Markov Control Processes

27

In particular, by (7.2), (7.4)

λ · ( µ, c

µ∗ , c ) > λ · u

∀ µ ∈ Mδ (IK), u ∈ B,

and taking µ = µ∗ we obtain that λ · u < 0 for all u ∈ B. Therefore, choosing an arbitrary i ∈ {1, . . . , q} and letting u ∈ B be the vector with components ui = −1 and uj = 0 for j = i, we conclude that λi > 0; hence, as i ∈ {1, . . . , q} was arbitrary, λ 0. Thus to complete the proof it only remains to verify that µ∗ and λ satisfy (7.1), so that µ∗ indeed minimizes λ · µ, c = µ, cλ . Suppose that this is not the case and let µ ∈ Mδ (IK) be such that µ, cλ < µ∗ , cλ , i.e. µ − µ∗ , cλ < 0.

(7.5)

For each r ≥ 0, let vr be the vector in T0 defined as vr := r( µ, c

µ∗ , c ) = r µ − µ∗ , c .

as r → ∞, which conThen, by (7.5), λ · vr = r µ − µ∗ , cλ tradicts (7.4). This completes the proof of Theorem 3.4 when π ∗ is a proper Pareto policy.

Let us now suppose that π ∗ is a weak Pareto policy and let µ∗ be the corresponding occupation measure. Then µ∗ , c is a weak Pareto point of Γ(Π0 ), i.e. there is no µ ∈ Mδ (IK) such that µ, c µ∗ , c .

Let C1 := {u ∈ IRq |u

µ∗ , c },

C2 := {u ∈ IRq |u

µ, c for some µ ∈ Mδ (IK)}.

Then C1 and C2 are disjoint convex sets, and in addition C1 is open.

Therefore, by the separation theorem in [32, p. 133, Theorem 3], there is a q–vector λ = 0 and a real number α such that (7.6)

λ·u<α≤λ·v

∀ u ∈ C1 , v ∈ C2 .

Moreover, the vector µ∗ , c is in the intersection of C2 and the closure of C1 , which yields that α = λ · µ∗ , c . Hence the first inequality in (7.6) gives λ · ( µ∗ , c w) ≤ λ · µ∗ , c w ∈ IRq+ ,

which implies that λ · w ≥ 0 for all w ≥ 0, and so λ ≥ 0. Thus λ > 0 because λ = 0. Finally, by the definition of C2 and the second inequality in (7.6), we obtain that λ · µ∗ , c λ · µ, c for all µ ∈ Mδ (IK), which concludes the proof of Theorem 3.4.

28

O. Hern´andez-Lerma and R. Romera

8

Further remarks

In this final section we briefly discuss some connections between our results and other problems for MCPs. Constrained MCPs. For each i = 1, . . . , q, let Vi (π) = Vi (π, γ0 ) be as in (2.4), and let k2 , . . . , kq be q − 1 nonnegative given numbers.

Then the problem minimize V1 (π) (8.1)

subject to: Vi (π) ≤ ki for i = 2, . . . , q; π ∈ Π,

is called a constrained MCP. In this case, a policy π for which (8.1) holds and, in addition, V1 (π) < ∞ is said to be feasible for the constrained MCP. Let us suppose that the set Πco ⊂ Π of feasible policies is nonempty. Then, under Assumption 3.1, there is an optimal policy π ∗ ∈ Πco for the constrained MCP (see e.g. [16]), and under an additional Slater–like condition, π ∗ is also a Pareto policy for the multiobjective MCP in Section 2 above; see [30], for instance.

For additional results on constrained MCPs or for MCPs with weighted criteria, see, for instance, [1, 10, 11, 14, 16, 17, 28, 30, 33, 38].

Average cost. Let us rewrite (2.4) as n−1

(8.2)

Vi (π, γ0 ) = lim sup Eγπ0 n→∞

n−1

δt ci (xt , at ) / t=0

δt . t=0

This is, of course, the same as (2.4) if 0 < δ < 1, whereas if δ = 1 we get the average cost (AC) criterion (8.3)

1 Ji (π, γ0 ) = lim sup Eγπ0 n→∞ n

n−1

ci (xt , at ) . t=0

It is easily verified that all of the results in Sections 3, 4 and 5 remain valid when δ = 1, with some obvious changes. For example, the set M1 (IK) in Definition 4.1 (and (4.5)) is the set of probability measures µ on X × A, concentrated on IK, and such that (as in (4.4)) (8.4)

µ(B) =

Q(B|x, a)µ(d(x, a)).

IK

Multiobjective Markov Control Processes

Similarly, by (8.4), the constraint equation (6.9) in the multiobjective LP formulation becomes (8.5)

L1 µ = 0, µ ∈ M+ (IK),

where L1 is given by (6.6) with δ = 1. Finally, as in the discounted case (8.1), we can also consider constrained MCPs with the AC criterion and obtain an optimal policy for the constrained problem, which is a Pareto policy for the multiobjective MCP. For details see [17], where a probability measure µ for which (8.5) holds is called stable.

Mixed average-discounted criteria. The average cost case in (8.3)-(8.5) can be used to study multiobjective MCPs with cost vectors of the form (J1 (π, γ0 ), . . . , Jr (π, γ0 ), Vr+1 (π, γ0 ), . . . , Vq (π, γ0 )) in which the Ji (π, γ0 ) are ACs as in (8.3), and the Vj (π, γ0 ) are discounted costs as in (8.2) with possibly different discount factors δj (j = r + 1, . . . , q). The key fact that allows us to do this is that the original multiobjective MCP is reduced to solving a Pareto problem of the form (4.7) but on the set M1 (IK) of stable probability measures. The corresponding technical details are essentially the same as in Remarks 2.2(c) and 3.7(b) of [17].

Further research: the balance space approach. In this paper we used two main approaches to analyze a multiobjective MCP: the scalarization approach (to study the problem of existence of Pareto policies) and the MMP approach (to study the characterization of Pareto policies). In fact, the former approach is the “dual” of the latter in a precise sense (see Section 6). On the other hand, there is a nonscalarized approach called the balance space approach introduced by Galperin [13] for vector optimization problems. This approach, in addition to allowing an interesting economic interpretation of the so–called “balance points”, has proved to be very effective from the computational viewpoint and also to study key issues, such as the sensitivity of vector minimization problems [6]. It might be worth investigating if this effectiveness also holds for multiobjective control problems.

29

30

O. Hern´andez-Lerma and R. Romera

On´esimo Hern´ andez-Lerma Departamento de Matem´ aticas, CINVESTAV-IPN, A.P. 14-470, M´exico D.F. 07000, M´exico. ohernand@math.cinvestav.mx

Rosario Romera Departamento de Estad´ıstica, Universidad Carlos III de Madrid Calle Madrid 126, Getafe 28903, Madrid Espa˜ na, rosario.romera@uc3m.es

References [1] Altman E., Constrained Markov Decision Processes, Chapman & Hall /CRC, Boca Raton, FL, 1999.

[2] Anderson E.J.; Nash P., Linear Programming in InfiniteDimensional Spaces, Wiley, Chichester, U.K., 1987.

[3] Arrow K.J.; Barankin E.W.; Blackwell D.,Admissible points of convex sets, Annals of Mathematics Studies, 2 (1950), pp. 87–91.

[4] Aubin J.-P.,A Pareto minimum principle, in Differential Games and Related Topics (Proc. Internat. Summer School, Varenna), North-Holland, Amsterdam, 1970, pp. 147-175.

[5] Aubin J.-P., Optima and Equilibria, Springer-Verlag, Berlin, 1993.

[6] Balb´ as A.; Guerra P.J., Measuring the balance space sensitivity in vector optimization, Lecture Notes Economics and Math. Syst., to appear.

[7] Balb´ as A.; Heras A., Duality theory for infinite dimensional multiobjective linear programming, Euro. J. Oper. Res., 68 (1993), 379388.

[8] Billingsley P., Convergence of Probability Measures, Wiley, New York, 1968.

[9] Dynkin E.B.; Yushkevich A.A., Controlled Markov Processes, Springer-Verlag, Berlin, 1979.

[10] Feinberg E.; Shwartz A., Constrained discounted dynamic programming, Math. Oper. Res., 21 (1996), 922-945.

[11] Feinberg E.; Shwartz A., Constrained dynamic programming with two discount factors: applications and an algorithm, IEEE Trans.

Autom. Control, 44 (1999), 628-631.

Multiobjective Markov Control Processes

31

[12] Furukawa N., Characterization of optimal policies in vector-valued Markov decision processes, Math. Oper. Res., 5 (1980), 271-279.

[13] Galperin E.A., Pareto analysis vis–` a–vis balance space approach in multiobjective global optimization, J. Optim. Theory Appl., 93 (1997), 533-545.

[14] Ghosh M.K., Markov decision processes with multiple costs, Oper.

Res. Lett., 9 (1990), 257-260.

[15] Gonz´alez-Hern´andez J.; Hern´andez-Lerma O., Envelopes of sets of measures, tightness, and Markov control processes, Appl. Math.

Optim., 40 (1999), 377-392.

[16] Hern´andez-Lerma O.; Gonz´alez-Hern´andez J., Constrained Markov control processes in Borel spaces: the discounted case, Math. Meth.

Oper. Res., 52 (2000), 271-285.

[17] Hern´andez-Lerma O.; Gonz´alez-Hern´andez J.; L´opez-Mart´ınez R.R., Constrained average cost Markov control processes in Borel spaces, SIAM J. Control Optim., 42 (2003), 442-468.

[18] Hern´andez-Lerma O.; Hoyos-Reyes L.F., A multiobjective control approach to priority queues, Math. Meth. Oper. Res., 53 (2001), 265–277.

[19] Hern´andez-Lerma O; Hoyos-Reyes L.F., A multiobjective formulation of optimal control problems with additive costs, Internal Report 286, Departamento de Matem´aticas, CINVESTAV–IPN, 2000.

[20] Hern´andez-Lerma O; Lasserre J.B., Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer-Verlag, New York, 1996.

[21] Hern´andez-Lerma O; Lasserre J.B., Further Topics on DiscreteTime Markov Control Processes, Springer-Verlag, New York, 1999.

[22] Hern´andez-Lerma O.; Vega-Amaya O.; Carrasco G., Sample-path optimality and variance-minimization of average cost Markov control processes, SIAM J. Control Optim., 38 (2000), 79-93.

[23] Henig M.I., Vector-valued dynamic programming, SIAM J. Control Optim., 21 (1983), 490-499.

32

O. Hern´andez-Lerma and R. Romera

[24] Henig M.I., The principle of optimality in dynamic programming with returns in partially ordered sets, Math. Oper. Res., 10 (1985), 462-471.

[25] Hilgert N.; Hern´andez-Lerma O., Bias optimality versus strong 0discount optimality in Markov control processes with unbounded costs, Acta Appl. Math., 77 (2003), 215-235.

[26] Hordijk A.; Yushkevich A.A., Blackwell optimality in the class of all policies in Markov decision chains with a Borel state space and unbounded rewards, Math. Meth. Oper. Res., 50 (1999), 421-448.

[27] Jahn J., Theory of vector maximization: various concepts of efficient solutions, Internat. Ser. Oper. Res. Management Sci., 21 (1999). Chapter 2.

[28] Krass D.; Filar J.; Sinha S.S, A weighted Markov decision process, Oper. Res., 40 (1992), 1180-1187.

[29] Lai H.-C.; Tanaka K., Average-time criterion for vector-valued Markovian decision systems, Nihonkai Math. J., 2 (1991), 71-91.

[30] L´opez-Mart´ınez R.R.; Hern´andez-Lerma O., The Lagrange and Pareto approaches to constrained Markov control processes, Morfismos, 7 (2003), 1-26.

[31] Luc D.T., Theory of Vector Optimization, Lecture Notes in Economics and Math. Systs. 319, Springer–Verlag, Berlin, 1989.

[32] Luenberger D.G., Optimization by Vector Space Methods, Wiley, New York, 1969.

[33] Piunovskiy A.B., Optimal Control of Random Sequences in Problems with Constraints, Kluwer, Boston, 1997.

[34] Reid R.W.; Citron S.J., On noninferior performance index vectors, J. Optim. Theory Appl., 7 (1971), 11-28.

[35] Salukvadze M.E., Vector-Valued Optimization Problems in Control Theory, Academic Press, New York, 1979.

[36] Sawaragi Y.; Nakayama H.; Tanino T., Theory of Multiobjective Optimization, Academic Press, New York, 1985.

Multiobjective Markov Control Processes

33

[37] Tanaka K., The closest solution to the shadow minimum of a cooperative dynamic game, Computers Math. Appl., 18 (1989), 181-188.

[38] Tanaka K., On discounted dynamic programming with constraints, J. Math. Anal. Appl., 155 (1991), 264-277.

[39] Tanaka K.; Matsuda C., On continuously discounted vector valued Markov decision process, J. Inform. Optim. Sci., 11 (1990), 33-48.

[40] Thomas L.C., Constrained Markov decision processes as multiobjective problems, Inst. Math. Appl. Conf. Ser., (1983), 77-94.

[41] Wakuta K., Vector-valued Markov decision process and the systems of linear inequalities, Stoch. Proc. Appl., 56 (1995), 159-169.

[42] Wakuta K.; Togawa K., Solution procedures for multi-objective Markov decision processes, Optimization, 43 (1998), 29-46.

[43] Waltz F.M., An engineering approach: hierarchical optimization criteria, IEEE Trans. Autom. Control, 12 (1967), 179-180.

[44] White C.C., III; Kwang W.K, Solution procedures for vector criterion Markov decision processes, Large Scale Systems, 1 (1980), 129-140.

[45] White D.J., Multi-objective infinite-horizon discounted Markov decision processes, J. Math. Anal. Appl., 89 (1982), 639-647.

[46] Zadeh L.A., Optimality and non-scalar-valued performance criteria, IEEE Trans. Autom. Control, 8 (1963), 59-60.

Morfismos, Vol. 8, No. 1, 2004, pp. 35–55

Tutte uniqueness of locally grid graphs D. Garijo

A. M´arquez

∗

M.P. Revuelta

Abstract A graph is said to be locally grid if the structure around each of its vertices is a 3 × 3 grid. As a follow up of the research initiated in [4] and [3] we prove that most locally grid graphs are uniquely determined by their Tutte polynomial.

2000 Mathematics Subject Classification: 05C75, 05C10.

Keywords and phrases: Locally grid graph; Tutte polynomial.

1

Introduction

Given a graph G, the Tutte polynomial of G is a two-variable polynomial T (G; x, y), which contains considerable information on G [1]. A graph G is said to be Tutte unique if T (G; x, y) = T (H; x, y) implies G ∼ =H for every other graph H. In Section 2 we prove that, locally grid graphs are Tutte unique.

Given a fixed graph H, a connected graph G is said to be locally H if for every vertex x the subgraph induced on the set of neighbors of x is isomorphic to H. For example, if P is the Petersen graph, then there are three locally P graphs [2]. The locally grid condition is slightly different since it involves not only a vertex and its neighbors, but also four vertices at distance two. From now on, all graphs considered have no isolated vertices.

We first recall some definitions and results about locally grid graphs from [4]. ∗ Invited article. Partially supported by projects BFM2001-2474-ORI and PAI FQM-164. This work is part of the first author’s Ph.D. thesis writen at the University of Sevilla.

35

36

D. Garijo, A. M´arquez and M.P. Revuelta

Let N (x) be the set of neighbors of a vertex x. We say that a 4−regular connected graph G is a locally grid graph if for each vertex x there exists an ordering x1 , x2 , x3 , x4 of N (x) and four different vertices y1 , y2 , y3 , y4 , such that, taking the indices modulo 4, N (xi ) ∩ N (xi+1 )

=

{x, yi }

N (xi ) ∩ N (xi+2 )

=

{x}

and there are no more adjacencies among {x, x1 , . . . , x4 , y1 , . . . , y4 } than those required by these conditions (Figure 1).

Figure 1: Locally Grid Structure Locally grid graphs are simple, two-connected, triangle-free, and each vertex belongs to exactly four cycles of length 4.

Let H = Pp × Pq be the p × q grid, where Pl is a path with l vertices.

Label the vertices of H with the elements of the abelian group Zp × Zq in the natural way. Vertices of degree four already have the locally grid property, hence we have to add edges between vertices of degree two and three in order to obtain a locally grid graph. A complete classification of locally grid graphs is given in [4], and they fall into the following families. In all the Figures, the vertices of the graph are represented by dots and two points with the same label correspond to points that are identified in the surface. δ with p ≥ 5, 0 ≤ δ ≤ p/2, δ + q ≥ 5 if q ≥ 4, The Torus Tp,q δ + q ≥ 6 if q = 2, 3 or 4 ≤ δ < p/2 with δ = p/3, p/4 if q = 1.

(Figure 2a) δ ) E(Tp,q

=

E(H)

∪

{{(i, 0), (i + δ, q − 1)}, 0 ≤ i ≤ p − 1}

∪

{{(0, j), (p − 1, j)}, 0 ≤ j ≤ q − 1}.

For δ = 0 we obtain the toroidal grid Cp × Cq , in this case we will write Tp,q . We can assume that δ ≤ p/2.

1 with p ≥ 5, p odd, q ≥ 5. (Figure 2b) The Klein Bottle Kp,q 1 ) E(Kp,q

=

E(H)

∪

{{(j, 0), (p − j − 1, q − 1)}, 0 ≤ j ≤ p − 1}

∪

{{(0, j), (p − 1, j)}, 0 ≤ j ≤ q − 1}.

37

Tutte uniqueness of locally grid graphs

11

12

6

7

8

9

10

12

11

10

9

8

7

6

5

5

5

5

4

4

4

4

3

3

3

3

2

2

2

2

1

1

1

1

6

7

8

9

10

11

12

6

7

8

9

a)

10

11

12

b)

2 b) K 1 Figure 2: a) T7,5 7,5

0 with p ≥ 6, p even, q ≥ 4 (Figure 3a).

The Klein Bottle Kp,q 0 ) E(Kp,q

=

E(H)

∪

{{(j, 0), (p − j − 1, q − 1)}, 0 ≤ j ≤ p − 1}

∪

{{(0, j), (p − 1, j)}, 0 ≤ j ≤ q − 1}.

2 with p ≥ 6, p even, q ≥ 5 (Figure 3b).

The Klein Bottle Kp,q 2 ) E(Kp,q

11

=

10

E(H)

9

8

7

∪

{{(j, 0), (p − j, q − 1)}, 0 ≤ j ≤ p − 1}

∪

{{(0, j), (p − 1, j)}, 0 ≤ j ≤ q − 1}.

6

6

11

10

9

8

7

5

5

5

5

4

4

4

4

3

3

3

3

2

2

2

2

1

1

1

1

6

7

8

9

10

11

6

7

a)

8

9

10

11

b)

0 b) K 2 Figure 3: a) K6,5 6,5

The graphs Sp,q with p ≥ 3 and q ≥ 6. (Figure 4). If p ≤ q

38

D. Garijo, A. M´arquez and M.P. Revuelta

E(Sp,q )

=

E(H)

∪

{{(j, 0), (p − j, q − p + j)}, 0 ≤ j ≤ p − 1}

∪

{{(0, i), (i, q − 1)}, 0 ≤ i ≤ p − 1}

∪

{{(0, i), (p − 1, i − p)}, p ≤ i ≤ q − 1}.

∪

{{(j, 0), (0, q − 1 − j)}, 0 ≤ j ≤ q − 1}

∪

{{(p − 1 − i, q − 1), (p − 1, i)}, 0 ≤ i ≤ q − 1}

∪

{{(i, q − 1), (i + q, 0)}, 0 ≤ i ≤ p − q − 1}.

If q ≤ p E(Sp,q )

6

7

E(H)

=

8

9

5

10

4

3 2 1

6

7

8 9

5 13

10 11

4

12

12 3

13

11 2 9

10 1

10

9 8

13

11 5

7

12

4

12

3 11 10 9 8 7 6

6

1

2

3

4

5

2

13

1

10 9 8 7 6

6

7

8

b)

a)

Figure 4: a) S5,8 b) S8,5 Theorem 1.1 [4] If G is a locally grid graph with N vertices, then exactly one of the following holds: δ with pq = N , p ≥ 5, δ ≤ p/2 and δ + q ≥ 5 if q ≥ 4 or a) G ∼ = Tp,q δ + q ≥ 6 if q = 2, 3 or 4 ≤ δ < p/2, δ = p/3, p/4 if q = 1.

Tutte uniqueness of locally grid graphs

39

i with pq = N , p ≥ 5, i ≡ p (mod 2) for i ∈ {0, 1, 2} and b) G ∼ = Kp,q q ≥ 4 + i/2 .

c) G ∼ = Sp,q with pq = N , p ≥ 3 and q ≥ 6.

2

Tutte Uniqueness

Let G = (V, E) be a graph with vertex set V and edge set E. The rank of a subset A ⊆ E is defined by r(A) = |A| − k(A), where k(A) is the number of connected components of the spanning subgraph (V, A). The rank-size generating polynomial is defined as: xr(A) y |A|

R(G; x, y) = A⊆E

The coefficient of xi y j in R(G; x, y) is the number of spanning subgraphs in G with rank i and j edges. This polynomial contains exactly the same information about G as the Tutte polynomial, which is given by: T (G; x, y) = A⊆E

(x − 1)r(E)−r(A) (y − 1)|A|−r(A)

hence, the Tutte polynomial tells us for every i and j the number of edgesets in G with rank i and size j. This fact is going to be essential in order to prove the Tutte uniqueness of locally grid graphs. Given a locally grid graph G, we show that for every locally grid graph H different from G and with |V (G)| = |V (H)| there is at least one coefficient of the rank-size generating polynomial in which both graphs differ.

Let S be the surface in which a locally grid graph G is embedded, that is, S is a torus or a Klein bottle [4]. Given two cycles C and C of G, we say that C is locally homotopic to C if there exists a cycle of length four, say H, with C ∩ H connected and C is obtained from C by replacing C − (C ∩ H) with H − (C ∩ H). A homotopy is a sequence of local homotopies. A cycle of G is called essential if it is not homotopic to a cycle of length four.

Let lG be the minimum length of an essential cycle of G. Note that lG is invariant under isomorphism. The number of essential cycles of length lG contributes to the coefficient alG −1,lG of R(G; x, y), which counts the number of edges sets with rank lG − 1 and size lG .

In order to show the Tutte uniqueness of locally grid graphs we are going to use the following results proved in [4]:

40

D. Garijo, A. M´arquez and M.P. Revuelta

Lemma 2.1 [4] Given two graphs G and G , if G is locally grid and T (G; x, y) = T (G ; x, y) then G is locally grid.

Lemma 2.2 [4] Let G, G be a pair of locally grid graphs with pq vertices then: a) lG = lG implies T (G; x, y) = T (G ; x, y). b) If lG = lG but G and G do not have the same number of shortest essential cycles, then T (G; x, y) = T (G ; x, y).

The process we are going to follow is to pairwise compare all the graphs given in the classification theorem of locally grid graphs. In those cases for which the minimum length of essential cycles or the number of cycles of this minimum length are different we have that both graphs are not Tutte equivalent, thus the relevance of the following result.

Lemma 2.3 If G is a locally grid graph with pq vertices, then the length lG of the shortest essential cycles and the number of these cycles are given in the following table: G Tp,q

δ Tp,q

lG

number of essential cycles

min{p, q}

min{p, q + δ}

if p < q

2p p q

if p = q if p > q

q+δ−1 δ

q+p p

q

q+δ−1 δ q

if p < q + δ if p = q + δ if p > q + δ if p < q + 1

0 Kp,q

min{p, q + 1}

5q 4q q

1 Kp,q

min{p, q}

q+1

if p = q

1 q

if p > q if p < q

q+2

if p = q

2

if p > q

2 Kp,q

min{p, q} 2p

Sp,q

min{2p, q}

if p = q + 1 if p > q + 1 if p < q

q−1 j

p−1 j=0

2p − 1 p

q(q − p) 2q

if p ≤ q ≤ 2p if 2p ≤ q if q ≤ p

Tutte uniqueness of locally grid graphs

41

δ and K i are proved in [4], where it is also Proof: The cases Tp,q , Tp,q p,q shown that lSp,q = min(2p, q) and that if q ≤ p the number of shortest essential cycles is 2q . Hence, we are only left with two cases in which we are given lower bounds on the number of essential cycles of length lSp,q . We are interested in calculating the exact number.

Locally grid graphs with pq vertices are constructed by adding edges to the p × q grid. These edges are called exterior edges. Essential cycles of shortest length are obtained by joining the two ends of an exterior edge by a path contained in the grid p × q. In Sp,q we distinguish two cases.

Case 1 If 2p ≤ q, every exterior edge of the form {(0, i), (p−1, i−p)} 2p − 1 essential cycles of length 2p. We have q−p edges determines p of this kind and each of them can use up to q different vertices, therefore 2p − 1 the number of essential cycles of length 2p is q(q − p) . p Case 2 If p ≤ q ≤ 2p, {(0, i), (i, q − 1)} and {(i, 0), (p − 1, q − p + i)} q with 0 ≤ i ≤ p − 1 generate essential cycles of length q. These i edges can use up to p different vertices, hence the number of essential q−1 . cycles of length q is 2p p−1 j=0 j

Theorem 2.4 Let p, q ≥ 6 verify the following conditions: q+δ−1 a) p = 2n for n ∈ N. δ b) pq = p q for all p , q ≥ 6 with p = q + δ = q + δ < p and p−1 p−1 =p . q+p δ δ δ is Tutte unique for all δ ≤ p/2.

Then Tp,q δ , x, y). By Proof: Let p, q ≥ 6 and G be a graph with T (G; x, y) = T (Tp,q Lemma 2.1, G is a locally grid graph, hence G has to be isomorphic to exactly one of the following graphs: Tp ,q , Tpδ ,q , Kpi ,q , Sp ,q . We prove that G is isomorphic to Tpδ ,q with p = p , q = q and δ = δ assuming that G is isomorphic to each one of the previous graphs and obtaining a contradiction in all the cases except in the aforementioned case. In [4] Tp,q was shown to be Tutte unique, thus we can consider δ > 0 and G not isomorphic to Tp ,q .

42

D. Garijo, A. M´arquez and M.P. Revuelta

Case 1 Suppose G ∼ = lK 0 δ = Kp0 ,q . By Lemma 2.2, lTp,q

p ,q

and the

number of shortest essential cycles has to be the same in both graphs.

Case 1.1 lTp,q = p, lK 0 = p with p < q + δ and p < q + 1. δ p ,q

As a result of Lemma 2.2, p = p and q = q . Our aim is to prove that the number of edge sets with rank q and size q + 1 is different for each graph. This would lead to a contradiction since this number is the coefficient of xq y q+1 in the rank-size generating polynomial. δ has k essential cycles of length q (δ > 1) or k + pq (δ = 1), If Tp,q 0 then Kp,q would have k + 4q such cycles. Therefore if we can show that there exits a bijection between edge sets with rank q and size q + 1 that are not essential cycles, we would have proved what we want.

For every r with 0 ≤ r ≤ q − 2 denote by Er the set {((i, r), (i, r + 1)) ; 0 ≤ i ≤ p − 1}. Let A be an edge set that is not an essential cycle δ . Define s(A) as min{r ∈ [0, q − 2] ; with rank q and size q + 1 in Tp,q δ ) the minimum always exits. For every r with A∩Er = ∅}. If A ⊂ E(Tp,q δ )|r(A) = 0 ≤ r ≤ q − 2 we define the bijection, ϕr between {A ⊆ E(Tp,q 0 q, |A| = q +1, s(A) = r} and {A ⊆ E(Kp,q )|r(A) = q, |A| = q +1, s(A) = r} as follows: δ ), ϕ (A) = ∪{ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ A} If A ⊂ E(Tp,q r where ⎧ j = q − 1, j = 0 (h, k) if ⎪ ⎪ ⎨ (h, k) if j, j ∈ [0, r] ψ((h, k)) = ⎪ ⎪ ⎩ (h, k) if r + 1 ≤ j, j ≤ q − 1 with h = (i, j), k = (i , j ), h = (p−1−i+δ, j) and k = (p−1−i +δ, j ). = p, lK 0 = p = q + 1 with p < q + δ or lTp,q = Case 1.2 lTp,q δ δ p ,q

q + δ < p, lK 0

p ,q

= p with p < q + 1.

The contradiction in these two cases is produced due to the equality of shortest essential cycles, number of these cycles and number of vertices on each graph.

Case 1.3 lTp,q = p, lK 0 = q + 1 with p < q + δ and q + 1 < p . δ p ,q

To obtain a contradiction, we are going to prove that there are more δ than in K 0 . Basically, edge-sets with rank q + 2 and size q + 3 in Tp,q p,q we are going to follow the same procedure that was developed in [4].

The previous sets can be classified into three groups: 1.- Normal edge-sets (they are edge-sets that do not contain any essential cycle).

Tutte uniqueness of locally grid graphs

43

2.- Sets containing an essential cycle of length q + 1 and two other edges (Figure 5a).

3.- Essential cycles of length q + 3 (Figures 5b and 6).

A B

B

A

A

A a)

b)

Figure 5: a) A set of edges in Kp0 ,q containing an essential cycle of length q + 1 and two other edges. b) Essential cycles of length q + 3 in δ .

Tp,q A

A

A

A

Figure 6: Essential cycles of length q + 3 in Kp0 ,q δ and K 0 (1) By Corollary 16 of [4] we know that Tp,q p ,q have the same number of normal edge-sets with rank q + 2 and size q + 3 that do not contain a cycle of length four. We are going to prove that the number of normal edge-sets with rank q + 2 and size q + 3 containing δ than in K 0 . a cycle of length four is greater in Tp,q p ,q Again by Corollary 16 of [4], the number of edge-sets with rank q + 1 and size q + 2 containing a cycle of length four is the same in both graphs, call it sq +1 . Add one edge to each of these sets in order to obtain a set with rank q + 2 and size q + 3. This set can be one of the following types depending on which edge we are adding: (a) A normal edge set with rank q + 2. (b) A normal edge set containing two non essential cycles and having rank q + 1. (c) An edge set containing an essential cycle of length q + 1 and a non essential cycle of length four. δ or K 0 ), the Let A(G), B(G) and C(G) (where G is either Tp,q p ,q number of edge-sets in G that belong to the groups A, B and C respec-

44

D. Garijo, A. M´arquez and M.P. Revuelta

tively. We recall the following equality from [4]: sq +1 (2pq − q − 2) = A(G)(q − 1) +

B∈B(G)

(q + 3 − δ(B)) + C(G)(q − 1)

where δ(B) is the number of edges of B which do not belong to any δ ) = 0 and C(K 0 ) = 0 we have: cycle of length four in B. Since C(Tp,q p ,q δ )(q − 1) + A(Tp,q

A(Kp0 ,q )(q − 1) +

δ ) (q B∈B(Tp,q

+ 3 − δ(B)) =

+ 3 − δ(B)) + C(Kp0 ,q )(q − 1).

B∈B(Kp0 ,q ) (q

Applying Corollary 16 several times we get that: (q + 3 − δ(B)) =

δ ) B∈B(Tp,q

B∈B(Kp0 ,q

(q + 3 − δ(B))

)

hence δ )(q − 1) = A(Kp0 ,q )(q − 1) + C(Kp0 ,q )(q − 1).

A(Tp,q δ , every essential cycle of length q + 1 plus two edges has (2) In Tp,q rank q + 2, but in Kp0 ,q there are essential cycles for which if we add two edges we obtain sets with rank q + 1. By hypothesis, both graphs have the same number of shortest essential cycles therefore the number δ than in K 0 . of edge-sets in this case is greater in Tp,q p ,q δ we have (3) For every essential cycle of length p = q + 1 in Tp,q p 2 ways of adding two edges in order to obtain a new essential 2 p δ there are 2q cycle, hence in Tp,q essential cycles of length q + 3.

2 q q +2 In [4] it is proved that in Kp0 ,q there are 4q +4 3 2 essential cycles of length q + 3. Since p = q + 1 and q = 4q , the δ than in number of essential cycles of length q + 3 is greater in Tp,q 0 Kp ,q .

Case 1.4 lTp,q = p = q + δ, lK 0 = p with q + 1 > p . δ p ,q

Suppose p = q + δ = p then q = q , hence δ < 1. We get a contradiction because δ ≥ 1. = p = q + δ, lK 0 = p = q + 1.

Case 1.5 lTp,q δ p ,q

45

Tutte uniqueness of locally grid graphs

If the length of the shortest essential cycles and the number of these cycles coincide in both graphs, we would have p = p , q = q , δ = 1 and q + pq = 5q therefore p = 4.

Case 1.6 lTp,q = p = q + δ, lK 0 = q + 1 with p > q + 1. δ p ,q

q +1=p=q+δ p > 4, q =

q q −1

q δ

<

Case 1.7 lTp,q = q + δ < p, lK 0 δ

p ,q

q +1=p =q+δ

and

⇒

4q = q + p

⇒

4q < p

q δ

q δ

,

.

= p = q + 1.

5q = p

q+δ−1 δ

.

If δ = 1 then q = q , p = p = q + δ < p hence δ > 1. p > 5 and q+δ−1 δ

=

q δ

⇒

>q

Case 1.8 lTp,q = q + δ < p, lK 0 δ

p ,q

5q < p

q+δ−1 . δ

=q +1<p.

Now, q + 1 = q + δ, so we can assume that δ > 1 because if δ = 1, then q = q , p = p and the number of shortest essential cycles would not be the same in both graphs. The contradiction in this case is similar q to the one obtained in the previous case because 4q = p . δ δ is not isomorphic After these eight cases we can conclude that Tp,q 0 to Kp,q .

Case 2 Suppose G isomorphic to Tpδ ,q .

Case 2.1 lTp,q = p, lT δ = p with p < q + δ and p < q + δ . δ p ,q

As a result of Lemma 2.2, p = p and q = q . Suppose δ < δ, as in case 1.1 our purpose is to prove that the number of edge sets with rank δ has x essential q + δ − 1 and size q + δ is different in each graph. If Tp,q q+δ −1 , therefore if we show cycles of length q + δ , Tpδ ,q has x+ p δ that there exits a bijection between the edge sets of non essential cycles with rank q + δ − 1 and size q + δ , we would have proved what we

46

D. Garijo, A. M´arquez and M.P. Revuelta

want. For every r with 0 ≤ r ≤ q − 2 we define the following bijection, δ )|r(A) = q + δ − 1, |A| = q + δ , s(A) = r} and ϕr between {A ⊆ E(Tp,q δ {A ⊆ E(Tp,q )|r(A) = q + δ − 1, |A| = q + δ , s(A) = r}.

δ ), ϕ (A) = ∪{ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ A} If A ⊂ E(Tp,q r where ⎧ (h, k) if j = q − 1, j = 0 ⎪ ⎪ ⎨ ψ((h, k)) = (h, k) if j, j ∈ [0, r] ⎪ ⎪ ⎩ (h, k) if r + 1 ≤ j, j ≤ q − 1

with h = (i, j), k = (i , j ), h = (i + δ − δ , j) and k = i + δ − δ , j ). = p, lT δ = p = q + δ with p < q + δ.

Case 2.2 lTp,q δ p ,q

δ ; x, y) then p = p and q = q + Suppose T (Tpδ ,q ; x, y) = T (Tp,q

q +δ −1 δ

p

. Since pq = p q we obtain q = q , a contradiction.

= p, lT δ Case 2.3 lTp,q δ

= q + δ with p < q + δ and q + δ < p .

p ,q

q +δ =p

⇒

q=p

pq = p q

⇒

p

p−1 δ

q +δ −1 δ

=q =p−δ

q +δ −1 > p. This contradiction was obtained δ by having assumed that both graphs have the same Tutte polynomial.

Because of hypothesis 2, we have that the case lTp,q = p = q + δ, δ lT δ = q + δ with q + δ < p cannot occur. δ < p − 1 then p

p ,q

With an analogous process to the one followed in case 1.3 we prove that the number of edge-sets with rank q + δ + 1 and size q + δ + 2 are δ and T δ . Therefore, l = q + δ, lT δ = q + δ < p different in Tp,q δ Tp,q p ,q p ,q

is not possible.

The rest of the cases are analogous to the previous ones, hence just one case can occur, namely, lTp,q = p = q + δ, lT δ = p = q + δ , which δ p ,q

implies p = p , q = q and δ = δ . δ ; x, y) = T (K 1 ; x, y).

Kp1 ,q , then T (Tp,q Case 3 Suppose G p ,q Because of Lemma 2.3, we cannot have p > q .

Tutte uniqueness of locally grid graphs

= p < q + δ, lK 1 Case 3.1 lTp,q δ

p ,q

47

=p <q.

As in case 1.1 we have to obtain a bijection to prove that the number of edge-sets with rank q − 1 and size q are different in each graph. δ ), ϕ (A) = ∪{ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ A} If A ⊂ A(Tp,q r where ⎧ j = q − 1, j = 0 (h, k) if ⎪ ⎪ ⎨ (h, k) if j, j ∈ [0, r] ψ((h, k)) = ⎪ ⎪ ⎩ (h, k) if r + 1 ≤ j, j ≤ q − 1 with h = (i, j), k = (i , j ), h = (p−1−i+δ, j) and k = (p−1−i +δ, j ).

The rest of the cases cannot occur because the length of shortest essential cycles, the number of these cycles and the number of vertices do not coincide. We omit the proof for the sake of brevity.

The case G Kp2 ,q is similar to the previous ones, hence we just specify the bijection in the case p = p < q and q = q : δ ), ϕ (A) = ∪{ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ A} If A ⊂ A(Tp,q r where ⎧ (h, k) if j = q − 1, j = 0 ⎪ ⎪ ⎨ ψ((h, k)) = (h, k) if j, j ∈ [0, r] ⎪ ⎪ ⎩ (h, k) if r + 1 ≤ j, j ≤ q − 1

with h = (i, j), k = (i , j ), h = (p − i + δ, j) and k = (p − i + δ, j ).

Case 4 Finally, we are going to assume that G Sp ,q . For the = p < q + δ and lSp ,q = 2p ≤ q or lTp,q = p = q +δ cases for which lTp,q δ δ and lSp ,q = q with q ≤ p we have that the length of shortest essential cycles, the number of these cycles and the number of vertices in both graphs, cannot coincide. Therefore we obtain a contradiction, since G and Sp ,q do not have the same Tutte polynomial. = q + δ < p, lSp ,q = q with By hypothesis 1 we cannot have lTp,q δ q ≤p.

Case 4.1 lTp,q = p < q + δ, lSp ,q = q with p ≤ q ≤ 2p . δ

Given that p = q , q = p and the equality of the number of shortest essential cycles q ≥ 2q −1 we arrive to a contradiction, because: p ≥ 2q −1 ≥ 2p −1 ⇒ 2p ≥ 2p . = p < q + δ, lSp ,q = q with q ≤ p Case 4.2 lTp,q δ Using the same ideas as in case 1.3 we prove that the number of δ than in S edge-sets with rank q + 1 and size q + 2 is greater in Tp,q p ,q .

For the sake of brevity we only give a sketch of the proof. These sets

48

D. Garijo, A. M´arquez and M.P. Revuelta

are classified into three groups: normal edge-sets, sets containing an essential cycle of length q and two other edges and essential cycles of δ has more edge sets of each type than length q + 2. We prove that Tp,q Sp ,q . The ideas are similar to case 1.3, so we just mention the last p δ we have 2 type. In Tp,q ways of adding two edges in order to get 2 p essential cycles of length a new essential cycle, hence there are 2q 2 q + 2. In Sp ,q (Figure 7) there are exterior edges to which we can add q q −1 different ways. Since p = q we have two edges in + 2 2 δ than in S more essential cycles of length q + 2 in Tp,q p ,q .

B

A

B

B

A

B

A

A

a)

b)

Figure 7: a) Edge sets in Sp ,q with p ≥ q containing an essential cycle of length q and two other edges. b) Essential cycles of length q + 2 in Sp ,q . = p = q + δ, lSp ,q = 2p with q ≥ 2p .

Case 4.3 lTp,q δ

Given that 2p = p = q + δ, we have q = 2q. We will obtain a contradiction by assuming we have equality for the number of shortest essential cycles in both graphs. In this case and in the next ones we 2p − 1 2p − 1 are going to use the following property: < if n m n < m ≤ [(2p − 1)/2] = p − 1.

If

q+p

q (q −p )

q+δ−1 δ

2p − 1 p

= q+p

= q (q − p ) 2p − 1 δ

< q+p

2p − 1 p

then

2p − 1 p −1

= q+p

2p − 1 p

49

Tutte uniqueness of locally grid graphs

<q+q

2p − 1 p

<q

1+

2p − 1 p

< q (q − p )

2p − 1 . p

= p = q + δ, lSp ,q = q with p ≤ q ≤ 2p .

Case 4.4 lTp,q δ q = p = q + δ then p = q. Suppose q+δ−1 q+p δ

= 2p

p −1 j=0

q −1 . j

δ ≤ p/2 = q /2 ≤ p = q. If δ < q ≤ p − 1 then: q+δ−1 δ

q+p <q

1+

q −1 δ

q −1 δ

=q+p

≤ 2p

q −1 δ

1+

q −1 δ

=p +q < 2p

p −1 j=0

q −1 j

.

q −1 q −1 ≤ because [(q − 1)/2] = q − 1. The δ q−1 difference between this case and the previous one is that the last bound is obtained as follows:

If δ = q,

2p

1+

q −1 δ

≤ 2p

1+

q −1 q−1

< 2p

p −1 j=0

q −1 j

Case 4.5 lTp,q = q + δ < p, lSp ,q = 2p with 2p ≤ q . δ 2p = q + δ

and

2p − 1 δ

Since [(2p − 1)/2] = p − 1, q (q − p )

2p − 1 p

q (q − p )

2p − 1 p

≥ pq

=p

2p − 1 δ

.

2p − 1 , p 2p − 1 ≥ (2p q − p q ) p

2p − 1 δ

>p

≤

2p − 1 . δ

= q + δ < p, lSp ,q = q with p ≤ q ≤ 2p .

Case 4.6 lTp,q δ q =q+δ

and

p

q+δ−1 δ

= 2p

p −1 j=0

q −1 j

.

.

50

D. Garijo, A. M´arquez and M.P. Revuelta

We will get a contradiction if we prove that q −1 δ

<2

p −1 j=0

q −1 . j

q ≤ 2p ⇒ [(q − 1)/2] ≤ p − 1 then ∃j0 ∈ [0, p − 1],

q −1 δ

≤

q −1 [(q − 1)/2]

q −1 [(q − 1)/2]

<2

p −1 j=0

q −1 j0 q −1 j

.

0 is Tutte unique for all p, q ≥ 6.

Theorem 2.5 Kp,q 0 ; x, y). Due Proof: Let p, q ≥ 6 and G a graph with T (G; x, y) = T (Kp,q to Lemma 2.1 and Theorem 2.4, G has to be isomorphic to exactly one of the following graphs: Kpi ,q , Sp ,q . We are going to prove that G is isomorphic to Kp0 ,q with p = p , q = q .

Suppose G isomorphic to Kp0 ,q then lTp,q = lK 0 and the number δ p ,q of shortest essential cycles has to be the same in both graphs. We just = p < q + 1, lK 0 = q + 1 with have to study the case in which lKp,q 0 p ,q

p > q + 1. This is so because, if lKp,q = q + 1 < p and lK 0 0

p ,q

= p

with p < q + 1 the reasoning would be analogous and in these cases it is easy to verify that the number of vertices and the length of shortest essential cycles can not coincide in both graphs.

0 = p < q + 1, lK 0 = q + 1 with p > q + 1 we can show that If lKp,q p ,q

the number of edge-sets with rank q + 2 and size q + 3 is different in 0 and K 0 . We omit the proof because it uses the same arguments Kp,q p ,q as those in case 1.3.

Suppose G ∼ = Kp1 ,q . Since p is even and p odd, all the cases in which 0 is p and in K 1 the length of shortest essential cycles in Kp,q p ,q is p are proved. By Lemma 2.3 we know that the number of shortest essential 0 is always bigger than one, hence we obtain a contradiction cycles in Kp,q in all those cases in which the number of shortest essential cycles in Kp1 ,q is one. Therefore we just have to study two cases: lKp,q = q + 1 < p, 0 = q + 1 < p, lK 1 = p = q . In the first lK 1 = p < q and lKp,q 0 p ,q

p ,q

51

Tutte uniqueness of locally grid graphs

case we obtain a contradiction by proving that the number of edge-sets with rank p + 1 and size p + 2 is different in each graph (following the same reasoning as in case 1.3 of Theorem 2.4). In the second case we show that if p = q = q + 1, pq = p q , p are even and p is odd it must then be the case that q is even. By Lemmas 2.2 and 2.3, 0 ; x, y) = T (K 2 ; x, y) if p > q , therefore G is not isomorphic T (Kp,q p ,q to Kp2 ,q . Following the same reasoning as in case 1.1 of Theorem 2.4 we show that it cannot be that lKp,q = p < q + 1 and lK 2 = p < q .

0 p ,q

0 )|r(A) = q, |A| = We just specify the bijection between {A ⊆ E(Kp,q q + 1, s(A) = r} and {A ⊆ E(Kp2 ,q )|r(A) = q, |A| = q + 1, s(A) = r}.

0 ), ϕ (A) = ∪{ψ(((i, j), (i , j ))); ((i, j), (i , j )) ∈ A} If A ⊂ A(Kp,q r where ⎧ j = q − 1, j = 0 (h, k) if ⎪ ⎪ ⎨ (h, k) if j, j ∈ [0, r] ψ((h, k)) = ⎪ ⎪ ⎩ (h, k) if r + 1 ≤ j, j ≤ q − 1

with h = (i, j), k = (i , j ), h = (i + 1, j) and k = (i + 1, j ).

On the other hand, we prove (as in case 1.3 of Theorem 2.4) that if lKp,q = q + 1 < p and lK 2 = p < q the number of edge-sets of rank 0 p ,q

p + 1 and size p + 2 is different for each graph.

The other four cases obtained by considering the possible combina0 and K 2 , are tions of the lengths of shortest essential cycles in Kp,q p ,q not possible since the length of shortest essential cycles, the number of these cycles and the number of vertices cannot coincide in both graphs.

Finally, suppose G Sp ,q .

Case 1 If lKp,q 0 = p < q + 1, lSp ,q = q with p ≤ q ≤ 2p we obtain a contradiction as follows: p<q+1⇒p ≤q <p +1⇒p =q ⇒p=q =p =q . q ≥ 2q −1 = 2q−1 .

Case 2 lKp,q = p < q + 1 and lSp ,q = q ≤ p .

0

As we did in case 1.3 of Theorem 2.4 we show that there are different 0 and S number of edge-sets with rank q + 1 and size q + 2 in Kp,q p ,q , hence these graphs do not have the same Tutte polynomial.

0 = p = q + 1 and lSp ,q = 2p ≤ q .

Case 3 lKp,q

52

D. Garijo, A. M´arquez and M.P. Revuelta

2p − 1 p will obtain a contradiction if we prove that 5q < q (q − p ).

2p = p = q + 1 then q = 2q. Since 5q = q (q − p )

we

q (q − p ) = 2q(2q − (p/2)) = 4q 2 − pq = 4q 2 − q(q + 1) = q(3q − 1) > q5 = q + 1 < p and lSp ,q = 2p ≤ q .

Case 4 lKp,q 0 2p − 1 2p − 1 2p = q + 1. 4q = q (p − q ) ≥ 2p 2 > p p 8p > 8p − 4 = 4(2p − 1) = 4q.

0 and K 2 , the rest of Similarly as in the comparisons between Kp,q p ,q the cases cannot occur because the length of shortest essential cycles, the number of these cycles and the number of vertices do not coincide in both graphs given that q ≥ 6.

1 is Tutte unique for all p, q ≥ 6.

Theorem 2.6 The graph Kp,q

Proof: The argument of this proof is basically the same as those followed δ and in Theorems 2.4 and 2.5. Because of the Tutte uniqueness of Tp,q 0 we only have to prove that T (G; x, y) = T (K 1 ; x, y) with G ∈ Kp,q p,q 1 2 {Kp ,q (except if p = p and q = q ), Kp ,q , Sp ,q }. In every case we are

1 ; x, y) and we will obtain a going to suppose that T (G; x, y) = T (Kp,q contradiction.

Case 1 If G Kp1 ,q it is easy to prove that the length of shortest essential cycles, the number of these cycles and the number of vertices only coincide if p = p and q = q .

Case 2 If G Kp2 ,q , by Lemmas 2.2 and 2.3 we get a contradiction in all those cases for which the number of shortest essential cycles in 1 is one or the number of shortest essential cycles in K 2 Kp,q p ,q is two.

In the other cases a contradiction is reached because p is odd and p is even.

Case 3 If G Sp ,q we can consider p ≤ q because if p > q the 1 is one and p, q ≥ 6. number of shortest essential cycles in Kp,q If lKp,q = p < q and lSp ,q = 2p ≤ q or lSp ,q = q with p ≤ q ≤ 2p , 1

by Lemmas 2.2 and 2.3 it is easy to obtain a contradiction. The same is true if lKp,q 1 = p = q and lSp ,q = q with q ≤ p or lSp ,q = q with

p ≤ q ≤ 2p .

1 = p < q and lSp ,q = q ≤ p we prove (as in previous If lKp,q cases) that there are different number of edge-sets with rank q + 1 and

53

Tutte uniqueness of locally grid graphs

size q + 2 in both graphs, hence they can not have the same Tutte polynomial.

If lKp,q = p = q and lSp ,q = 2p ≤ q , 2p = p = q then q = 2q 1 therefore q + 1 = q (q − p )

2p − 1 p

= 2q(q − p )

2p − 1 p

> q + 1.

2 is Tutte unique for p, q ≥ 6.

Theorem 2.7 The graph Kp,q

Proof: Due to Theorems 2.4, 2.5 and 2.6 we have to prove that T (G; x, y) 2 ; x, y) with G ∈ {K 2 = T (Kp,q p ,q (except if p = p and q = q ), Sp ,q } and pq = p q .

2 ; x, y) if p = p and

By Lemmas 2.2 and 2.3, T (Kp2 ,q ; x, y) = T (Kp,q q = q because the length of shortest essential cycles, the number of these cycles and the number of vertices only coincide if p = p and q=q.

2 has If G Sp ,q we can assume that p ≤ q otherwise if q < p, Kp,q two shortest essential cycles and by Lemma 2.2 we obtain a contradiction. = p < q and lSp ,q = q ≤ p we prove as in previous Case 1 If lKp,q 2 cases that the number of edge-sets with rank q + 1 and size q + 2 is different in both graphs. Hence these two graphs cannot have the same Tutte polynomial, therefore we get a contradiction and G can not be isomorphic to Sp ,q .

Case 2 If lKp,q 2 = p = q and lSp ,q = 2p ≤ q then 2p = p = q and q = 2q hence q + 2 = q (q − p )

2p − 1 p

= 2q(q − p )

2p − 1 p

> q + 2.

In the other four cases we obtain a contradiction because the length of shortest essential cycles, the number of these cycles and the number of vertices cannot coincide in both graphs.

Theorem 2.8 The graph Sp,q is Tutte unique for p, q ≥ 6 and 2q = q +δ−1 p for all p , q with p q = pq and δ > 0. δ Proof: Suppose that Sp,q is not Tutte unique. Then, by Theorems 2.4, 2.5, 2.6, 2.7 and Lemma 2.2 Sp,q is isomorphic to Sp ,q with p = p, q = q and pq = p q .

54

D. Garijo, A. M´arquez and M.P. Revuelta

Case 1 If lSp,q = 2p ≤ q and lSp ,q = q with p ≤ q ≤ 2p , by Lemma 2.2 q = 2p and q = 2p then: q(q − p)

2p − 1 p

> (2p − (q /2))

q −1 q /2 q −1 ≥ (2p − p ) q /2

= 2p (2p − (q /2)) q −1 q /2

q −1 = p q /2

>

p −1 j=0

q −1 . j

Hence, the number of shortest essential cycles is different in each graph and by Lemma 2.2 we have that Sp,q is not isomorphic to Sp ,q .

Case 2 If lSp,q = 2p ≤ q and lSp ,q = q ≤ p then 2p = q and q = 2p .

2p−1 2p − 1 2p − 1 q 2p 2 =2 =2 < 2 · 2p j p j=0

≤q

2p − 1 p

< q(q − p)

2p − 1 . p

We obtain a contradiction to the assumption that the number of shortest essential cycles is equal in both graphs.

The other three cases are analogous to the previous ones.

3

Concluding Remarks

We have shown that locally grid graphs are Tutte unique for p, q ≥ 6, but our techniques do not apply to p = 3, 4, 5. An interesting open q +δ−1 problem is to prove that the number p is not a power of δ two. This would give a more general result about the Tutte uniqueness δ and S . of Tp,q p,q Acknowledgement We are very grateful to I. Gitler of CINVESTAV-I.P.N. for carefully reviewing the manuscript and for making useful suggestions.

Tutte uniqueness of locally grid graphs

D. Garijo Department of Applied Math., University of Sevilla, Avenida Reina Mercedes S/N, Sevilla C. P. 41012, Espa˜ na. dgarijo@us.es

A. M´ arquez Department of Applied Math., University of Sevilla, Avenida Reina Mercedes S/N, Sevilla C. P. 41012, Espa˜ na almar@us.es

M. P. Revuelta Department of Applied Math., University of Sevilla, Avenida Reina Mercedes S/N, Sevilla C. P. 41012, Espa˜ na pastora@us.es

References [1] Brylawsky T.; Oxley J., The Tutte polynomial and its applications, Matroid Applications, Cambridge University Press, Cambridge, 1992.

[2] Hall J.I., Locally Petersen graphs, J. Graph Theory, 4 (1980), 173187.

[3] M´arquez de Mier A.; Noy M., On graphs determined by their Tutte polynomials, Graphs Combin., 20 (2004), 105-119.

[4] M´arquez de Mier A.; Noy M.; Revuelta M.P., Locally grid graphs: Classification and Tutte uniqueness, Discr. Math., 266 (2003), 327-352.

[5] Thomassen C., Tilings of the Torus and the Klein Bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math.

Soc., 323 (1991), 605-635.

55

Morfismos, Vol. 8, No. 1, 2004, pp. 57–83

No-inmersi´on de espacios lente Enrique Torres Giese

1

Resumen Con herramientas b´asicas como la sucesi´on espectral de Serre y los cuadrados de Steenrod se obtienen resultados de no-inmersi´on de espacios lente de dimensi´on 2n+1 y torsi´on 2m . En la situaci´on α(n) = 1, donde α(n) es el n´ umero de 1’s en la expansi´on binaria de n, el resultado es ´optimo.

2000 Mathematics Subject Clasification: 57R42.

Keywords and phrases: Inmersi´on de variedades, sucesi´ on espectral de Serre, cuadrados de Steenrod.

1

Introducci´ on

Un problema cl´ asico de la Topolog´ıa Diferencial es el de conocer cu´ando una variedad M admite una inmersi´on de manera ´optima en un espacio euclideano, es decir, conocer el m´ınimo entero k para el cual M admite una inmersi´on en Rk . Este es un problema a´ un abierto, pues el decidir cu´ando una variedad M admite una inmersi´on en una variedad N es en extremo complicado. Una primer contribuci´on en la soluci´on de este problema ocurri´o en 1944, cuando Whitney demostr´o que toda variedad compacta de dimensi´on n admite una inmersi´on en un espacio euclideano de dimensi´on 2n − 1. En otras palabras, Whitney acot´o superiormente la dimensi´on del espacio euclideano donde la variedad pudiera tener una inmersi´on de manera ´optima. En este trabajo analizaremos el problema de inmersi´on de espacios lente, de hecho concluiremos un resultado de no inmersi´on de tales espacios.

1

Becario Conacyt 165576. El contenido de este art´ıculo est´ a basado en la tesis de Maestr´ıa presentada por el autor en el Departamento de Matem´ aticas del CINVESTAV-IPN.

57

58

Enrique Torres Giese

Si f : X n → Rn+k es una inmersi´on de la variedad X de dimensi´on n, entonces existe un haz νf de dimensi´on k tal que τX ⊕ νf es trivial.

El haz νf se nombra el haz normal asociado a la imersi´ on f . Cuando X es compacta Hirsch en [7] demostr´o el rec´ıproco: si existe un haz ν de dimensi´on k tal que τX ⊕ ν es trivial, entonces existe una inmersi´on X n → Rn+k cuyo haz normal asociado es ν. Observe que en el caso de tener una inmersi´on X n → Rn+k , el haz tangente τX es el inverso estable del haz normal νX .

El trabajo de Hirsch tambi´en afirma que entre cualesquiera dos inmersiones de X n en R2n+1 existe una homotop´ıa X × I → R2n+1 tal que cada X × {t} → R2n+1 es inmersi´on. Por lo que si f : X n → Rn+k f

y g : X n → Rn+l son inmersiones, entonces X n → Rn+k → R2n+1 g y X n → Rn+k → R2n+1 son homot´opicas, as´ı νf ⊕ (n − k + 1) = νg ⊕ (n − l + 1). Es decir, cualesquiera dos haces normales asociados a distintas inmersiones determinan la misma clase estable, llamada el haz normal estable de X y denotado por νX .

En estos t´erminos, el decidir si una variedad admite una inmersi´on en codimensi´on k es equivalente a conocer si la dimensi´on geom´etrica de νX , denotada por gd(νX ), es menor o igual que k, mientras que conocer la dimensi´on o´ptima de inmersi´on es equivalente a conocer exactamente gd(νX ). A su vez, el problema de encontrar la dimensi´on geom´etrica de un haz α es equivalente al problema de levantamiento BO(k)

X

α

BO

La idea clave en la demostraci´on del resultado principal, que a continuaci´on enunciamos, se basa en este u ´ ltimo hecho. En nuestro caso supondremos una inmersi´on, lo cual producir´a un levantamiento de la funci´on clasificante del haz normal y algebraicamente se probar´a su inexistencia.

Escribimos X ⊆ Y si la variedad X admite una inmersi´on en la variedad Y . Para n y m enteros positivos denotamos por L2n+1 (2m ) al espacio de ´orbitas asociado a la acci´on usual de Z/2m sobre los vectores de norma uno en Cn+1 .

Teorema Sea m ≥ 2 y l(n) = max{1 ≤ i ≤ n − 1 :

n+i+1 n

≡ 0 (4)}.

No-inmersi´on de espacios lente

59

a) Si n = 2s + 1 y n ≥ 2, entonces L2n+1 (2m ) ⊆ R2n+1+2l(n) .

b) Si n = 2s + 1, con s ≥ 1, entonces L2n+1 (2m ) ⊆ R2n+2l(n) = R4n−4 .

Corolario Sea m ≥ 2 a) Si n = 2s con s ≥ 1, entonces L2n+1 (2m ) ⊆ R4n−1 .

b) Si n = 2s + 2t con s > t ≥ 1, entonces L2n+1 (2m ) ⊆ R4n−3 .

En la Subsecci´on 3.2 comparamos estos resultados con situaciones conocidas de inmersi´on de espacios complejos proyectivos.

2

Preliminares

La sucesi´ on de Gysin es una sucesi´on en cohomolog´ıa asociada a un p i haz esf´erico S k → E → B orientable. Esta sucesi´on se obtiene a partir de la sucesi´on exacta larga de la pareja (D, E) (con E → D y D → B el haz de discos asociado) y el isomorfismo de Thom produciendo φ

p∗

·e

· · · → H r−1 (E) → H r−k−1 (B) → H r (B) → H r (E) → · · · donde e ∈ H k+1 (B) es la clase de Euler del haz esf´erico. La condici´on de que el haz sea orientable puede cubrirse si suponemos que su base sea simplemente conexa. p

i

Lema 2.1 Si en el haz esf´erico S k → E → B, B es 1-conexo y su clase de Euler es cero, entonces H ∗ (E; Z) ∼ = H ∗ (B; Z) ⊕ a · H ∗ (B; Z) odulo, donde a ∈ H k (E; Z) es tal que φ(a) es un como un H ∗ (B; Z)-m´ generador de H 0 (B; Z).

Demostraci´ on: Es importante realizar dos comentarios. Primero, que la forma en que H ∗ (E) es visto como H ∗ (B)-m´odulo es a trav´es de p.

Segundo, que el morfismo φ es la composici´on del morfismo de conexi´on y el isomorfimo de Thom, de modo que φ es un morfismo de H ∗ (B)m´odulos. Observe que la sucesi´on de Gysin toma la forma p∗

φ

0 → H q (B) → H q (E) → H q−k (B) → 0.

60

Enrique Torres Giese

Sea a ∈ H k (E) tal que φ(a) = 1 ∈ H 0 (B) ∼ = Z, entonces el morfismo ∗ ∗ H (B) → H (E) dado por b → b · a escinde esta u ´ ltima sucesi´on, obteniendo as´ı el resultado buscado.

Por otra parte, en el estudio de haces vectoriales, las clases de StiefelWhitney y de Chern juegan un papel esencial. En nuestro estudio manejaremos otros elementos que surgen de la complejificaci´on de haces reales. Estos elementos en cohomolog´ıa son llamadas las clases de Pontrjagin. Estas se definen para haces reales como pi (ξ) = (−1)i c2i (ξ C ) ∈ H 4i (X) con n = dimR (ξ), i ≤ [ n2 ] y cj (ξ C ) la j-´esima clase de Chern de ξ C . En particular, las clases de Pontrjagin del haz universal λn → BO(n) (o de su versi´on universal λ+ n → BSO(n)) se llaman clases universales de Pontrjagin y se denotan por pk . En la Observaci´on 2.2 estaremos de hecho interesados en las clases de λ+ 2n cuya clase de Euler denotaremos por en (ver Observaci´on 2.3 (a)).

Observaci´ on 2.2 A continuaci´on mencionamos algunas propiedades y relaciones de las clases de Pontrjagin y la clase de Euler. a) La funci´on U (n) → SO(2n) A + iB →

A −B B A

induce una funci´on rn : BU (n) → BSO(2n) la cual satisface que rn∗ (λ+ onicos 2n ) = r(γn ). Este hecho junto con los isomorfismos can´ ξ C = ξ C , r(ξ C ) = ξ ⊕ ξ y r(η)C = η ⊕ η¯ nos permiten obtener las relaciones rn∗ (pk ) = c2k − 2ck−1 · ck+1 + . . . + (−1)k+1 2c2k−1 · c1 + (−1)k 2c2k rn∗ (en ) = cn

b) Si ξ es un haz orientable con dim(ξ) = n, entonces pn (ξ) = e(ξ)2 .

Observaci´ on 2.3 Algunas observaciones u ´ tiles en las siguientes secciones: a) Tanto S 2n−1 → BU (n − 1) → BU (n) y S n−1 → BSO(n − 1) → BSO(n) son respectivamente los haces esf´ericos de los haces can´onicos γn → BU (n) y λn → BSO(n) [15], lo cual nos permite considerar sus respectivas clases de Chern, Pontrjagin y Euler.

61

No-inmersi´on de espacios lente

i

b) Puesto que la funci´on BU (n) → BU (m) induce al haz universal γm → BU (m) en γn ⊕ (m − n), entonces i∗ ci (γm ) = ci (γn ⊕ (m − n)) = ci (γn ) por lo que la funci´on i∗ : H ∗ (BU (m)) → H ∗ (BU (n)) es un isomorfismo hasta dimensi´on 2n. Similarmente, i la funci´on BSO(n) → BSO(m) induce un isomorfismo en cohomolog´ıa m´odulo 2 hasta dimensi´on n.

Lema 2.4 Si n − k es par, entonces H n−k (Vn,k ) ∼ = Z. Demostraci´ on:

Considere la fibraci´on i

S n−k → Vn,k → Vn,k−1 .

Puesto que Vn,k y Vn,k−1 son (n − k − 1) y (n − k) conexos respectivamente, podemos aplicar la sucesi´on exacta de Serre en cohomolog´ıa. En particular observe que tenemos la siguiente sucesi´on exacta ∗

i H n−k (Vn,k−1 ) → H n−k (Vn,k ) → H n−k (S n−k ) ∼ =Z

donde H n−k (Vn,k−1 ) = 0. Veamos que en el caso n − k par el morfismo i∗ es un isomorfismo. Para ello veamos que el morfismo δ de conexi´on siguiente es trivial. El morfismo δ se encuentra inducido por la diferencial dn−k+1 de la sucesi´on espectral de Serre y est´a definido como multiplicaci´on por la clase de Euler de la fibraci´on en cuesti´on. Ahora, considere el siguiente diagrama de fibraciones S n−k

S n−k ∗

i

Vn−k+2

S n−k+1

Vn,k

Vn,k−1

Vn,k−2

Vn,k−2

La sucesi´on espectral de Serre del segundo rengl´on se mapea en la del primer rengl´on, de hecho utilizando la sucesi´on exacta de Serre se verifica que el morfismo H n−k+1 (Vn,k−1 ) → H n−k+1 (S n−k+1 ) es un isomorfismo lo cual, en vista de la naturalidad de la clase de Euler, implica que la clase de Euler buscada es trivial. As´ı, H n−k (Vn,k ) ∼ = Z. α

Lema 2.5 Sean Vn,k → BSO(n − k) la funci´ on clasificante del haz i

SO(n − k) → SO(n) → Vn,k y S n−k = Vn−k+1,1 → Vn,k la inclusi´ on natural. Si n − k es par, entonces α∗ (e) = ±2u donde u es un generador de H n−k (Vn,k ) y e ∈ BSO(n − k) es la clase de Euler.

62

Enrique Torres Giese

Demostraci´ on:

Considere el siguiente diagrama

S n−k−1

Vn−k+1,2

S n−k i

S n−k−1

Vn,k+1

Vn,k α

S n−k−1

BSO(n − k − 1)

BSO(n − k)

BSO(n)

BSO(n)

donde el segundo rengl´on es el haz esf´erico can´onico sobre BSO(n − k) inducido por α y el primer rengl´on el haz inducido por i. Observe que el haz S n−k−1 → Vn−k+1,2 → S n−k es el haz esf´erico del haz tangente de S n−k el cual, cuando n − k es par, tiene por clase de Euler a 2 ∈ H n−k (S n−k ). Escribimos a α∗ (e) como ku, donde k ∈ Z y u es un generador de H n−k (Vn,k ). Como i∗ es un isomorfismo en este caso (Lema 2.4) y la clase de Euler es natural, entonces k = ±2.

Finalmente, recordamos que la cohomolog´ıa de los espacios lente de torsi´on 2m est´a dada por H ∗ (L2n+1 (2m ); Z/2) ∼ = E(x) ⊗ Pn+1 (z), con |x| = 1 y |z| = 2.

3

El espacio clasificante de haces reales establemente complejos El material de esta secci´on est´a basado en el trabajo de B. Junod [9].

3.1

La cohomolog´ıa de B(n, k)

En esta primer subsecci´on estudiaremos las propiedades cohomol´ogicas de un espacio particular que nos permitir´a desarrollar nuestro trabajo.

Definici´ on 3.1 Consid´erese el siguiente diagrama BSO(k)

BU (n)

BSO(2n)

No-inmersi´on de espacios lente

63

Decimos que el pull-back de este sistema es el espacio clasificante de haces reales de dimensi´ on k establemente complejos sobre complejos celulares de dimensi´ on ≤ 2n + 1 y lo denotamos por B(n, k). Observe que inductivamente podemos definir el espacio B(n, k) como el pull-back del diagrama BSO(k)

B(n, k + 1)

BSO(k + 1)

Observaci´ on 3.2 En la definci´on anterior se debe cumplir que 1 ≤ k ≤ 2n, para el caso en que k = 2n se verifica que B(n, 2n) = BU (n). Por otra parte, si k = 2n − 1 y P es el pullback del sistema SO(2n − 1) U (n)

SO(2n)

donde U (n) → SO(2n) est´a dada por A −B B A

A + iB →

entonces existe una funci´on U (n − 1) → P inducida por las funciones U (n−1) → U (n) y U (n−1) → SO(2n−2) → SO(2n−1). De hecho, tal situaci´on forza a que U (n−1) → P sea un isomorfismo. Este argumento muestra que B(n, 2n − 1) = BU (n − 1).

Trabajaremos frecuentemente con el siguiente diagrama V2n,2n−2j

V2n,2n−2j

i1

B(n, 2j)

i2 f2j

BSO(2j)

p

BU (n)

rn

BSO(2n)

64

Enrique Torres Giese

Observe que existe una funci´on h : BU (j) → B(n, 2j) determinada por la propiedad universal de B(n, 2j) BU (j) rj

h

i

B(n, 2j)

f2j

BSO(2j)

p

BU (n)

rn

BSO(2n)

Lema 3.3 Para cada n ≥ 1 y 1 ≤ j ≤ n − 1, existe un elemento aj ∈ H 2j (B(n, 2j); Z) tal que ∗ (e ) = p∗ (c ) − 2a , i∗ (a ) = u , h∗ (a ) = 0 f2j j j j j j 1 j

donde ej ∈ H 2j (BSO(2j); Z) es la clase universal de Euler y uj ∈ H 2j (V2n,2n−2j ; Z) es un generador.

Demostraci´ on:

De acuerdo al Lema 2.5 podemos elegir uj ∈ H 2j (V2n,2n−2j ; Z) ∼ =Z

generador tal que i2 (ej ) = −2uj . Puesto que V2n,2n−2j es (2j − 1)conexo, BU (n) es 1-conexo y la cohomolog´ıa de BU (n) est´a concentrada en dimensiones pares, aplicando la sucesi´on exacta larga de Serre a la fibraci´on V2n,2n−2j → B(n, 2j) → BU (n) se tiene la sucesi´on exacta corta 0 → H 2j (BU (n)) → H 2j (B(n, 2j)) → H 2j (V2n,2n−2j ) → 0 Sea x ∈ H 2j (B(n, 2j)) tal que i∗1 (x) = uj . Puesto que i∗ es un isomorfismo hasta dimensi´on 2j podemos reemplazar a x por aj = x − p∗ (i∗ )−1 h∗ (x), as´ı h∗ (aj ) = 0 y i∗1 (aj ) = uj . Por otra parte, como H 2j (V2n,2n−2j ; Z) ∼ = Z, entonces la sucesi´on exacta anterior se escinde produciendo el isomorfismo H 2j (B(n, 2j)) ∼ = im(p∗ ) ⊕ Zaj ∼ = H 2j (BU (n)) ⊕ Zaj ∗ (e ) = r ∗ (e ) = c , entonces f ∗ (e ) = p∗ (c )+ma . Adem´ Como h∗ f2j as, j j j j j j 2j j ∗ ∗ ∗ ∗ ∗ i1 f2j (ej ) = i2 (ej ) = −2uj , as´ı m = −2 y f2j (ej ) = p (cj ) − 2aj .

65

No-inmersi´on de espacios lente

Una situaci´on particular de nuestro diagrama de trabajo es la siguiente S 2j−1 S 2j−1 B(n, 2j − 1)

f2j−1

BSO(2j − 1)

p2j−1

B(n, 2j)

f2j

BSO(2j)

Considere la sucesi´on de Gysin de la fibraci´on p2j−1

S 2j−1 → B(n, 2j − 1) → B(n, 2j) la cual tiene clase de Euler (de acuerdo al Lema anterior) p∗ (cj ) − 2aj → H q (B(n, 2j))

·p∗ (cj )−2aj

−→

H q+2j (B(n, 2j)) → H q+2j (B(n, 2j − 1)) →

y por exactitud se satisface que p∗2j−1 p∗ (cj ) = 2p∗2j−1 (aj ).

Denotaremos por bj a p∗2j−1 (aj ), m´as generalmente bj denotar´a a p∗k p∗k+1 · · · p∗2j−1 (aj ), mientras que cj al elemento (p∗k p∗k+1 · · · p∗2j−1)p∗ (cj ).

Con esta notaci´on se satisface que 2bj = cj para 1 ≤ j ≤ n − 1. La siguiente figura muestra tal situaci´on.

B(n, 2j − 1) ↓ B(n, 2j) ↓ B(n, 2n − 5) ↓ B(n, 2n − 4) ↓ B(n, 2n − 3) ↓ B(n, 2n − 2) ↓ B(n, 2n − 1) ↓ BU (n)

bj , bj+1 , . . . , bn−1 , c1 , . . . , cj−1

2bj = cj

aj , bj+1 , . . . , bn−1 , c1 , . . . , cj bn−2 , bn−1 , c1 , . . . , cn−3

2bn−2 = cn−2

an−2 , bn−1 , c1 , . . . , cn−2 bn−1 , c1 , . . . , cn−2 an−1 , c1 , . . . , cn−1 c1 , . . . , cn−1 c1 , . . . , cn

2bn−1 = cn−1

66

Enrique Torres Giese

Observaci´ on 3.4 Hacemos un par de observaciones u ´ tiles en la demostraci´on del siguiente resultado. ∗ (e ) = c − 2a en H 2k (B(k + a) De acuerdo al Lema anterior f2k k k k ∗ (e2 ) = 1, 2k)), y en vista de la Observaci´on 2.2 (ck − 2ak )2 = f2k k ∗ (p ) = p∗ r ∗ (p ) = p∗ (c2 − 2c 2 . Por lo tanto f2k c ) = c k k k−1 k+1 k+1 k k 4ck ak = 4a2k en H ∗ (B(k + 1, 2k).

B(k + 1, 2k)

ak , c1 , . . . , ck

BU (k) = B(k + 1, 2k + 1)

c1 , . . . , ck

BU (k + 1)

c1 , . . . , ck+1

b) Utilizando la fibraci´on V2n,2n−k → B(n, k) → BU (n) y en vista de que V2n,2n−k y BU (n) son 1-conexo, se tiene que B(n, k) es 1-conexo. c) Sea h : B(n − 1, 2j) → B(n, 2j) la funci´on que define la propiedad universal de B(n, 2j) y que es compatible con la misma funci´on h del diagrama previo al Lema 3.3. En esta situaci´on se tiene el diagrama B(n − 1, 2j)

h

B(n, 2j)

BSO(2j)

B(n − 1, 2n − 3)

B(n, 2n − 3)

BSO(2n − 3)

BU (n − 1)

B(n, 2n − 2)

BSO(2n − 2)

BU (n − 1)

B(n, 2n − 1)

BSO(2n − 1)

BU (n − 1)

BU (n)

BSO(2n)

el cual nos muestra que < bn−1 >⊆ Ker(h∗ ) ya que h∗ (an−1 ) = 0 (Lema 3.3).

67

No-inmersi´on de espacios lente

d) En el Lema 3.3 construimos elementos aj ∈ H 2j (B(n, 2j)), n´otese que estos elementos en principio dependen de n, para ser precisos aj deber´ıa denotarse como anj . Veamos que h preserva tales elementos, es decir h(anj ) = ajn−1 . Para esto considere el siguiente diagrama conmutativo V2n,2n−2j−2 i

V2n,2n−2j−2

B(n − 1, 2j)

BSO(2j) h

B(n, 2j)

BU (n − 1)

BSO(2j)

BSO(2n − 2) BU (n)

BSO(2n)

donde la funci´on i es la inclusi´on can´onica que es parte de la fii braci´on V2n,2n−2j−2 → V2n,2n−2j → V2n,2 . Utilizando la sucesi´on exacta de Serre se tiene que i∗ es un isomorfismo en dimensi´on 2j, lo cual implica que h∗ (aj ) al restringirlo en su correspondiente variedad de Stiefel es un generador, en particular h∗ (aj ) y aj son libres de torsi´on. La observaci´on es consecuencia de la conmutatividad del diagrama, del hecho de que h∗ (aj ) y aj son libres de torsi´on y de la relaci´ones del Lema 3.3 que definen dichos elementos. En adelante usaremos indistintamente la notaci´on aj para referirnos a los elementos del Lema 3.3 sin importar n. odulo libre determinado por el Teorema 3.5 H ∗ (B(n, k); Z) es un Z-m´ isomorfismo H ∗ (B(n, k); Z) ∼ =

Z[c1 , . . . , ct ] ⊗ ∆(at , bt+1 , . . . , bn−1 ) k = 2t Z[c1 , . . . , ct ] ⊗ ∆(bt+1 , . . . , bn−1 ) k = 2t + 1

68

Enrique Torres Giese

donde ∆(x1 , . . . , xm ) es el grupo abeliano libre generado por los elementos xi1 xi2 · · · xis , 1 ≤ i1 < i2 < · · · < is ≤ m.

Demostraci´ on: Procederemos por inducci´on decreciente sobre k. Los casos en que k es 2n ´ o 2n−1 son inmediatos pues H ∗ (B(n, k)) es en estos casos, de acuerdo a la Observaci´on 3.2, H ∗ (BU (n)) y H ∗ (BU (n − 1)) respectivamente. Analicemos ahora el caso k = 2n − 2 y para ´ello la sucesi´on de Gysin de la fibraci´on p2n−2

que es

S 2n−2 → B(n, 2n − 2) → B(n, 2n − 1) p∗2n−2

φ

0→H 2q (BU (n−1)) → H 2q (B(n, 2n−2)) → H 2q−2n+2 (BU (n−1))→ 0 pues su clase de Euler es trivial al ser de grado impar en BU (n − 1).

As´ı, de acuerdo al Lema 2.1, tenemos el isomorfismo H ∗ (B(n, 2n − 2)) ∼ = H ∗ (BU (n − 1)) ⊕ a · H ∗ (BU (n − 1)) para a ∈ H 2n−2 (B(n, 2n − 2)) tal que φ(a) es generador de H 0 (BU (n − 1)). Por otra parte, aplicando la sucesi´on exacta de Serre a la fibraci´on p V2n,2 → B(n, 2n − 2) → BU (n) se tiene la sucesi´on exacta corta p∗

0 → H 2n−2 (BU (n)) → H 2n−2 (B(n, 2n − 2)) → H 2n−2 (V2n,2 ) → 0 la cual produce el isomorfismo H 2n−2 (B(n, 2n − 2)) ∼ = Im(p∗ ) ⊕ Zan−1 2n−2 ∗ ∼ ya que H (V2n,2 ) = Z (Lema 2.4). Como Im(p ) = Im(p∗2n−2 ) = ker(φ), entonces φ(an−1 ) es generador de H 0 (BU (n − 1)), lo cual describe el isomorfismo deseado. Observe que en este caso, de acuerdo a la Observaci´on 3.4, a2n−1 = cn−1 an−1 .

Supongamos ahora el resultado cierto para k con r ≤ k ≤ 2n − 1 y prob´emoslo para r − 1.

Si r es impar, digamos r = 2j + 1, en este caso procedemos como arriba. Aplicamos la sucesi´on de Gysin a la fibraci´on S 2j → B(n, 2j) → B(n, 2j+1), la cual tiene clase Euler trivial, para concluir el isomorfismo H ∗ (B(n, 2j)) ∼ = H ∗ (B(n, 2j + 1)) ⊕ aj H ∗ (B(n, 2j + 1)) ∼ = Z[c1 , . . . , cj ] ⊗ ∆(aj , bj+1 , . . . , bn−1 )

69

No-inmersi´on de espacios lente

Para efectos de la demostraci´on del siguiente caso, en que r es par, demostraremos que el morfismo de grupos ψn

Z[c1 , . . . , cj−1 , cj − 2aj ] ⊗ ∆(aj , bj+1 , . . . , bn−1 ) → H ∗ (B(n, 2j)) x ⊗ y −→ xy

es un isomorfismo. Procedemos por inducci´on sobre n y para ´ello comenzamos en n = j + 1 que es el primer valor que puede tomar n. El morfismo en consideraci´on toma la forma ψj+1

Z[c1 , . . . , cj−1 , cj − 2aj ] ⊗ ∆(aj ) → Z[c1 , . . . , cj ] ⊗ ∆(aj ) y en vista de que a2j = aj cj las relaciones (cj − 2aj ) ⊗ 1 + 2(1 ⊗ aj ) → cj y (cj − 2aj )2 ⊗ 1 → c2j muestran que ψj+1 es suprayectiva y en consecuencia un isomorfimo entre Z-m´odulos libres. Supongamos que la afirmaci´on es cierta para valores menores que n y sean A = Z[c1 , . . . , cj ] ⊗ ∆(aj , bj+1 , . . . , bn−2 ), B = Z[c1 , . . . , cj−1 , cj − 2aj ] ⊗ ∆(aj , bj+1 , . . . , bn−2 ), {xi } base de A (por lo que {xbn−1 } es base de Abn−1 ) y h : B(n − 1, 2j) → B(n, 2j) la funci´on de la Observaci´on 3.4.

Observe por una parte que Ker(h∗ ) = Abn−1 y que H ∗ (B(n, 2j)) ∼ = A ⊕ Abn−1 . As´ı, la hip´otesis de inducci´on y la conmutatividad del diagrama B ψn |B

A ⊕ Abn−1

ψn−1 h∗

A

muestran que ψn |B es monomorfismo, y de hecho podemos elegir {yi } base de B de tal suerte que ψn (yi ) = xi + zi bn−1 para zi ∈ A. Puesto que a2n−1 = cn−1 an−1 y 2bn−1 = cn−1 , se cumple que b2n−1 = 0 en H ∗ (B(n, 2j)). Por lo tanto ψn |Bbn−1 es biyectiva y su imagen est´a en Abn−1 . As´ı, ψn es isomorfismo.

Si r es par, digamos r = 2j, entonces de acuerdo al Lema 3.3 la clase de Euler de la fibraci´on S 2j−1 → B(n, 2j −1) → B(n, 2j) es cj −2aj y en vista de que ψn es inyectiva, se cumple que el morfismo de multiplicaci´on por la clase de Euler es inyectivo, luego el morfismo φ de la sucesi´on de Gysin de la fibraci´on S 2j−1 → B(n, 2j − 1) → B(n, 2j) es trivial lo cual produce los isomorfismos de grupos H ∗ (B(n, 2j − 1)) ∼ = H ∗ (B(n, 2j))/ < cj − 2aj > ∼ = Z[c1 , . . . , cj−1 ] ⊗ ∆(bj , bj+1 , . . . , bn−1 ).

70

Enrique Torres Giese

Lema 3.6 Para cada n ≥ 1 y 1 ≤ j ≤ n − 1, el elemento aj ∈ H 2j (B(n, 2j); Z) satisface min(2j,n−1)

a2j = aj cj + (−1)j

(−1)r br c2j−r . r=j+1

Demostraci´ on: Recordemos que la clase de Euler ej ∈ H 2j (BSO(2j)) satisface la relaci´on e2j = pj y que min(2j,n)

rn∗ (pj )

=

c2j

j

(−1)r 2cr c2j−r

+ (−1)

r=j+1

en H 4j (BU (n)) (Observaci´on 2.2). As´ı, ∗ (e2j ) = c2j − 4aj cj + 4a2j f2j

y min(2j,n−1)

(−1)r 2cr c2j−r

rn∗ (pj ) = c2j + (−1)j

r=j+1 min(2j,n−1)

(−1)r 4br c2j−r .

= c2j + (−1)j r=j+1

El resultado es ya inmediato pues H ∗ (B(n, 2j)) es libre de torsi´on.

Observaci´ on 3.7 De nuevo hacemos tres observaciones u ´ tiles para la demostraci´on del siguiente resultado. a) La funci´on h∗ de la Observaci´on 3.4 define un isomorfismo hasta dimensi´on 2n − 3. (o < 2(n − 1)). b) Considere el siguiente diagrama B(n, 2j − 2) × CP ∞

f2j−2 ×1

B(n + 1, 2j)

BU (n) × CP ∞

BU (n + 1)

BSO(2j − 2)×CP ∞ f2j

BSO(2j)

BSO(2n + 2)

No-inmersi´on de espacios lente

71

donde el resto de las funciones son can´onicas. En esta situaci´on el cuadro externo es homot´opicamente conmutativo y en vista de que BSO(2j) → BSO(2n + 2) es una fibraci´on podemos sustituir la funci´on B(n, 2j −2)× CP ∞ → BSO(2j −2)× CP ∞ → BSO(2j) por una que haga tal cuadro extrictamente conmutativo. De esta manera existe una funci´on f de tal suerte que los cuadros B(n, 2j − 2) × CP ∞

BSO(2j − 2) × CP ∞

f

B(n + 1, 2j)

BSO(2j)

B(n, 2j − 2) × CP ∞

f

BU (n) × CP ∞

B(n + 1, 2j)

BU (n + 1)

sean (homot´opicamente) conmutativos.

Analizaremos cu´al es el efecto de f ∗ sobre H ∗ (B(n + 1, 2j)). Observe que la funci´on BU (n) × CP ∞ → BU (n + 1) muestra que c(γn × γ1 ) = c(πn∗ (γn ) ⊕ π1∗ (γ1 )) = c(πn∗ (γn ))c(π1∗ (γ1 ))

= (c(γn ) ⊗ 1) · (1 ⊗ c(γ1 )) = c(γn ) ⊗ c(γ1 )

donde πn : BU (n) × CP ∞ → BU (n) y π1 : BU (n) × CP ∞ → CP ∞ son las proyecciones can´onicas, as´ı f ∗(ci ) = ci + ci−1 z en H ∗ (B(n, 2j − 2) × CP ∞ ) ∼ = H ∗ (B(n, 2j − 2)) ⊗ H ∗ (CP ∞ ), donde ∗ z es el generador de H (CP ∞ ) y 1 ≤ i ≤ j.

Recordemos que cj = 2bj en B(n, 2j − 2). Ahora, puesto que ∗ (a ) = c − 2a y la funci´ fej on BSO(2j − 2) × CP ∞ → BSO(2j) j j j env´ıa e(λ2j ) en (e(λ2j−2 ) ⊗ 1) · (1 ⊗ z), se tiene que 2f ∗ (aj ) = cj + 2aj−1 y en consecuencia f ∗ (aj ) = bj + aj−1 z (pues H ∗ (B(n, 2j − 2) × CP ∞ ) es libre de torsi´on).

Por otra parte, la funci´on B(n, 2j − 2) × CP ∞ → B(n + 1, 2j) → B(n + 1, 2i) se factoriza B(n, 2j − 2) × CP ∞ → B(n, 2i − 2) × CP ∞ → B(n + 1, 2i) donde la u ´ ltima funci´on de esta composici´on es una versi´on de f para j = i. Utilizando este hecho tenemos que

72

Enrique Torres Giese

f ∗ (bi ) = bi + bi−1 z para j + 1 ≤ i ≤ n − 1 y que f ∗ (bn ) = bn−1 z pues bn = 0 en H ∗ (B(n, 2n − 2)) de acuerdo a la Observaci´on 3.4. c) Sea G = Z/2[c1 , . . . , cj−1 ] ⊗ (Z/2 < aj > ⊕Z/2 < bj+1 > · · · ⊕ Z/2 < bn >) → H ∗ (B(n+1), 2j); Z /2) y observe que la restricci´on de f ∗ a G es inyectiva.

Teorema 3.8 Para cada n ≥ 1, 1 ≤ j ≤ n − 1 y cada 0 ≤ k ≤ j se tiene la siguiente relaci´on en H ∗ (B(n, 2j); Z/2) k−1

Sq 2k (aj ) = r=max(0,k+j+1−n)

j−r bk+j−r cr + aj ck . k−r

Demostraci´ on: Procederemos por inducci´on sobre n. El caso en que n = 1 se verifica trivialmente, mientras que para n = 2 se tiene que j = 1 y k = 0, 1. Pero Sq 2 (a1 ) = a21 = a1 c1 lo cual coincide con nuestra relaci´on.

Suponga cierto el resultado para valores menores que n + 1. Para aj ∈ H 2j (B(n + 1, 2j)) se cumple que Sq 2k (aj ) ∈ H 2(k+j) (B(n + 1, 2j)), as´ı de acuerdo al isomorfismo de la Observaci´on 3.7 nuestra relaci´on es v´alida para k + j ≤ n − 1. Para k + j ≥ n consideraremos la situaci´on de la Observaci´on 3.7. Primero analicemos el caso en que j > 1, k < j y k + j ≥ n, de acuerdo a nuestra hip´otesis de inducci´on se tiene que h∗ Sq 2k (aj ) = Sq 2k (h∗ (aj )) = Sq 2k (aj ) k−1

= r=k+j+1−n

j−r bj+k−r cr + aj ck k−r

y puesto que ker(h∗ ) = bn H ∗ (B(n, 2j)) tenemos k−1

Sq 2k (aj ) = r=k+j+1−n

j−r bj+k−r cr + aj ck + bn p(c1 , . . . , cj+k−n ) k−r

donde p ∈ Z/2[c1 , . . . , cj−1 ]. Luego, Sq 2k (aj ) ∈ G y f ∗ Sq 2k (aj ) = Sq 2k (f ∗ (aj )) = Sq 2k (bj + aj−1 z) = Sq 2k (bj ) + Sq 2k (aj−1 )z + Sq 2k−2 (aj−1 )z 2 .

73

No-inmersi´on de espacios lente

Aplicando de nuevo la hip´otesis de inducci´on, vemos que f ∗ Sq 2k (aj ) est´a dado por k

k−1 j−r j−r−1 bj+k−r cr + bj+k−r−1cr z k−r k−r r=k+j−n

r=k+j+1−n k−2

j−r−1 bj+k−r−2 cr z 2 + (aj−1 ck z + aj−1 ck−1 z 2 ) k−r−1

+ r=max(0,k+j−n−1)

y como

j−r−1 k−r

k

≡

j−r−1 k−r−1

j−r k−r

+

(2), f ∗ Sq 2k (aj ) ahora es

k−1 j−r j−r bj+k−r (cr + cr−1 z) + bj+k−r−1 cr z k−r k − r r=k+j−n

r=k+j+1−n

k−1

j−r bj+k−r−1cr−1 z 2 + aj−1z(ck + ck−1 z) k−r

+ r=max{1,k+j−n}

Si k + j > n, se tiene k

f ∗ Sq 2k (aj ) = r=k+j+1−n k−1

+ r=k+j−n

j−r bj+k−r (cr + cr−1 z) k−r

j−r bj+k−r−1 z(cr + cr−1 z) + aj−1 z(ck + ck−1 z) k−r

k−1

= r=k+j+1−n

j−r (bj+k−r + bk+j−r−1 z)(cr + cr−1 z) k−r

+(bj + aj−1 z)(ck + ck−1 z) + ⎛

= f∗ ⎝

k−1 r=max(0,k+j−n)

n−k bn−1 z(cj+k−n + cj+k−n−1 z) n−j ⎞

j−r bk+j−r cr + aj ck ⎠ k−r

que es nuestro resultado ya que f ∗ |G es inyectiva. Si k + j = n, procedemos de manera an´aloga como arriba. Restan dos casos, el primero para j = 1 y el segundo para j = k. En el primer caso la relaci´on deseada se verifica f´acilmente, mientras que el segundo es consecuencia del Lema 3.6 y que Sq 2j (aj ) = a2j .

74

Enrique Torres Giese

3.2

El problema de inmersi´ on

A continuaci´on mencionamos c´omo un sistema de Moore-Postnikov est´a relacionado con el problema de levantamiento.

Proposici´ on 3.9 Sea g : W → Y con W un CW -complejo finito. Si f {Zn , αn , βn } es un sistema de Moore-Postnikov de F → X → Y , entonces una condici´ on suficiente para que g levante a trav´es de f es que H ∗ (W, π∗−1 (F )) = 0.

Una aplicaci´ on de este resultado la podemos hacer al siguiente problema. Cada haz vectorial complejo estable se clasifica a trav´es de una funci´on X → BU . Si X es un CW complejo finito, digamos de dimensi´on N , la pregunta es si podemos levantar la funci´on de clasificaci´on a alg´ un BU (n) y si es el caso c´omo depende n de N . N´otese que un tal levantamiento siempre es posible para n suficientemente grande pues X es compacto y BU est´a dotado con la topolog´ıa uni´on de los BU (n).

Puesto que la fibra W de BU (n) → BU es ∪k≥0 Wn+k,k y cada Wn+k,k es 2n-conexo, se tiene que W es 2n-conexo. As´ı, para q ≤ 2n + 1 tenemos H q (X, πq−1 (W )) = 0, y si suponemos que n ≥ 12 (N − 1) entonces H q (X, πq−1 (W )) = 0 para q ≥ 2n + 2, por lo que tendremos un levantamiento a BU (n).

Un razonamiento an´alogo al anterior muestra que si X es un CW complejo de dimensi´on N y existe un haz complejo ξ sobre X con funci´on clasificante X → BU (M ) y M > N , entonces la funci´on clasificante de ξ levanta a BU (n), para 2n + 1 ≥ N , produciendo un isomorfismo ξ = η ⊕ (M − n) para η un haz complejo sobre X.

En particular, cualquier haz complejo estable sobre L2n+1 (2m ) se clasifica por una funci´on L2n+1 (2m ) → BU (n). El trabajo de est´a secci´on puede ser interpretado como un m´etodo para detectar obstrucciones no triviales para levantar levantar haces L2n+1 (2m ) → BU (n) a alg´ un BU (m) con m < n.

Enunciamos de nuevo el resultado principal de este trabajo.

Teorema 3.10 Sea m ≥ 2 y l(n) = max{1 ≤ i ≤ n − 1 : 0 (4)}.

n+i+1 n

a) Si n = 2s + 1 y n ≥ 2, entonces L2n+1 (2m ) ⊆ R2n+1+2l(n) .

≡

No-inmersi´on de espacios lente

75

b) Si n = 2s + 1, con s ≥ 1, entonces L2n+1 (2m ) ⊆ R2n+2l(n) = R4n−4 .

Corolario 3.11 Sea m ≥ 2 a) Si n = 2s con s ≥ 1, entonces L2n+1 (2m ) ⊆ R4n−1 . b) Si n = 2s + 2t con s > t ≥ 1, entonces L2n+1 (2m ) ⊆ R4n−3 .

En la demostraci´on de este teorema utilizaremos los isomorfismos can´onicos τCP n ⊕ 1C = (n + 1)γn y τL2n+1 (2m ) ⊕ 1R = (n + 1)σ, donde σ = q ∗ (r(γn )) y q : L2n+1 (2m ) → CP n la proyecci´on can´onica. Observe que estos isomorfimos muestran que si gd(νCP n ) = M , entonces gd(νL2n+1 (2m ) ) ≤ M . Esto u ´ltimo quiere decir que si CP n ⊆ R2n+k , entonces L2n+1 (2m ) ⊆ R2n+1+k . Por otra parte, Milgram prob´o en [11] que CP n ⊆ R4n−α(n)+1 y que CP n ⊆ R4n−α(n) si n es impar o si α(n) = 1. As´ı, cuando α(n) = 1 tenemos que L2n+1 (2m ) ⊆ R4n lo cual implica que en este caso nuestro resultado es ´optimo.

Por otra parte, en [13] Sanderson y Schwarzenberger demostraron que CP n ⊆ R4n−2α(n)−1 y que CP n ⊆ R4n−2α(n)+ si α(n) = 1 o si n es par con α(n) ≡ 0 (4), donde = 0 si α(n) ≡ 1 (4), y = 1 si α(n) ≡ 2, 3 (4).

Algunos casos del trabajo de Sanderson-Schwarzenberger pueden obtenerse (o compararse a partir de cierto punto) con el Teorema 3.10, por ejemplo: a) Si α(n) = 1, de acuerdo a nuestro resultado CP n ⊆ R4n−2 el cual es ´optimo en vista de la inmersi´on de Milgram y coincide con la no-inmersi´on de Sanderson-Schwarzenberger. b) Para n = 2s + 1 nuestro resultado afirma que L2n+1 (2m ) ⊆ R4n−4 y en consecuencia CP n ⊆ R4n−5 que coincide tambi´en con el resultado de Sanderson-Schwarzenberger. c) Para n = 2r + 2s nuestro resultado afirma que L2n+1 (2m ) ⊆ R4n−3 y en consecuencia CP n ⊆ R4n−4 mientras que el resultado de Sanderson-Schwarzenberger afirma que CP n ⊆ R4n−3 . d) Para n = 2r + 2s + 1 nuestro resultado afirma que L2n+1 (2m ) ⊆ R4n−7 y en consecuencia CP n ⊆ R4n−8 mientras que el resultado de Sanderson-Schwarzenberger afirma que CP n ⊆ R4n−7 .

76

Enrique Torres Giese

Finalmente, Davis y Mahowald [1] demostraron que para α(n) = 2 y n impar CP n ⊆ R4n−3 y si n es par CP n ⊆ R4n−2 . Por lo que en la situaci´on α(n) = 2 nuestro resultado de no-inmersi´on dista a lo m´as en una unidad de la situaci´on o´ptima.

Observaci´ on 3.12 Haremos a continuaci´on algunas observaciones sobre l(n). Denotamos por ν(n) al exponente de 2 en n. Usaremos las identidades ν ab = α(b) + α(a − b) − α(a) y α(a − 1) = α(a) − 1 + ν(a). a) Como ν

2n n

b) Como ν

2n−1 n

= α(n), entonces l(n) = n − 1 para α(n) = 1. = α(n) − 1, entonces l(n) = n − 2 para α(n) = 2.

c) Como ν 2n−2 = α(n − 2) + 1 − ν(n), entonces l(n) < n − 3 para n n impar y α(n) ≥ 3.

Lema 3.13 Si n = 2s1 + · · · + 2sk con s1 > · · · > sk ≥ 0 y k ≥ 3, entonces l(n) = 2s1 + 2s2 − 2 − 2s3 − · · · − 2sk .

Demostraci´ on: Usaremos el hecho bien conocido de que ν ab es el n´ umero de acarreos que ocurren al realizar la resta de las expresiones binarias de a y b. Sea r = 2s1 + 2s2 − 2 − 2s3 − · · · − 2sk . Entonces n + r + 1 = 2s1 +1 + 2s2 +1 − 1, por lo que el m´aximo n´ umero de acarreos en la resta de n + r + 1 y n es precisamente uno, en la posici´on s1 , como se muestra en la figura (ya sea que s1 = s2 + 1 o s1 > s2 + 1) s1 + 1 s1 · · · s2 + 1 s2 · · · sk 1 0 0 1 1 1 0 1 1

n+r+1 n

Por lo que r ≤ l(n). Veamos que r es de hecho l(n). Analicemos, para ejemplificar, la diferencia (n + r + 2) − n. En este caso (n + r + 2) − n = 2s1 +1 + 2s2 +1 + 1 y en tal situaci´on la figura s1 + 1 s1 · · · s2 + 1 s2 · · · sk 1 0 1 0 0 1 0 1 1

n+r+2 n

muestra que hay un acarreo en la posici´on sk y otro, en el peor de los casos, en s2 . Consideremos en adelante la diferencia (n+r +k +1)−n =

No-inmersi´on de espacios lente

77

r + k + 1 para k ≥ 1 la cual, de acuerdo a nuestra definici´on de l(n), debe ser a lo mas n.

Supongamos que s1 = s2 +1 y no existen acarreos en (n+r+k+1)−n, entonces (n + r + k + 1) − n ≥ 2s1 +1 > n que es una contradicci´on. De existir s´olo un acarreo tal situaci´on forza a que n + r + k + 1 tenga 1 en la posici´on s2 (de lo contrario habr´ıa dos o m´as acarreos) s1 + 1 s1 s2 · · · sk 1 1 1 0 1 1 1

n+r+k+1 n

as´ı (n + r + k + 1) − n ≥ 2s1 +1 > n.

Supongamos que s1 > s2 + 1, entonces existe un acarreo en (n + r + k + 1) − n en la posici´on s1 s1 + 1 s1 · · · s2 + 1 s2 · · · sk 1 0 1 1 0 1 1

n+r+k+1 n

y de no existir m´as acarreos (n + r + k + 1) − n ≥ 2s1 + 2s2 +1 > n. Por lo tanto l(n) = r.

Esta descripci´on de l(n) para α(n) ≥ 3 nos dice que ⎧ ⎪ 2 (4) ⎪ ⎪ ⎨

si n ≡ 0 1 (4) si n ≡ 1 l(n) ≡ ⎪ 0 (4) si n ≡ 2 ⎪ ⎪ ⎩ 3 (4) si n ≡ 3

(4) (4) (4) (4)

Demostraci´ on del Teorema 3.10 Puesto que τL2n+1 (2m ) ⊕ 1 = (n + 1)σ, L2n+1 (2m ) ⊆ R2n+1+k si y s´olo si, se tiene un levantamiento BSO(k)

−(n + 1)σ) : L2n+1 (2m )

BSO

78

Enrique Torres Giese

Al inicio de esta secci´on se discuti´o la existencia de un levantamiento g de −(n + 1)(γ) a BU (n + 1), de modo que se tiene el siguiente diagrama BSO(k)

B(n + 1, k) f¯k

L2n+1 (2m )

g

p

BU (n + 1)

BSO(2n + 2)

Observe que g∗ (ci ) = ci (−(n + 1)γ) = c(−(n + 1)γ) = c(γ)−n−1 = (1 + z)−n−1 = Para i ≥

[ k2 ] +

−n−1 i i≥0

z i = (−1)i −n−1 i z. i

n+i n

z i pues

1 se tiene que p∗ (ci ) = 2bi ∈ H ∗ (B(n + 1, k)), luego

2f¯k∗ (bi ) = g ∗ (ci ) = (−1)i n+i n

Esta u ´ltima identidad implica que 2f¯k∗ (bi ) = 2(−1)i

n+i i m z (2 ). n es par y que

1 n+i i m z (2 ).

2 n

As´ı 1 n + i i m−1 z (2 ) f¯k∗ (bi ) = 2 n y dado que m ≥ 2 se obtiene en particular 1 n+i i f¯k∗ (bi ) = z (2).

2 n

(1)

Ahora, si k = 2i y ai ∈ H 2i (B(n + 1, 2i)) con i ≤ n, entonces f¯k∗ (ai ) = λi z i ∈ H 2i (L2n+1 ; Z) con λi ∈ Z/2m .

∗ a Como Sq 2 (ai ) = ibi+1 + ai c1 y Sq 2 (λi z i ) = iλi z i+1 , al aplicar f¯2i se tiene la siguiente identidad

Sq 2 (ai ) (2)

iλi ≡ (n + 1)λi + i

que es v´alida para i + 1 ≤ n.

1 n+i+1 2 n

(2)

79

No-inmersi´on de espacios lente

Similarmente, como Sq 4 (ai ) = 2i bi+2 + (i − 1)bi+1 c1 + ai c2 y ∗ a Sq 4 (a ) se tiene la siguiente idenSq 4 (λi z i ) = 2i λi z i+2 , al aplicar f¯2i i tidad m´odulo 2 (3)

i n+2 i 1 n+i+2 1 n+i+1 λi ≡ λi +(i − 1)(n + 1) + 2 2 n 2 2 n 2

que es v´alida para i + 2 ≤ n.

Tomemos en adelante i = l(n) y m ≥ 2, por lo que (1) para i + 1, y en consecuencia (2) y (3), es v´alida, en caso de existir f¯2i . De hecho, de acuerdo a la definici´on de l(n), el tercer sumando de la ecuaci´on (3) es nulo, por lo que ´esta se transforma en (4)

i λi ≡ 2

n+2 1 n+i+1 λi + (i − 1)(n + 1) 2 2 n

(2).

Si n = 2s con s ≥ 1, en este caso tenemos que i = l(n) = n − 1, entonces la ecuaci´ on (2) nos dice que 1 n+i+1 2 n

≡ 0 (2)

lo cual contradice la definici´on de l(n).

Si n es par y α(n) ≥ 2, entonces (ver los comentarios siguientes al Lema 3.13 y (b) de la Observaci´on 3.12) i = l(n) es par y ≤ n − 2.

Entonces, de acuerdo a la ecuaci´on (2), tenemos que λi ≡ 0 (2), y en consecuencia la ecuaci´on (4) nos dice 0≡

1 n+i+1 2 n

(2)

lo cual contradice la definici´on de l(n).

Si n es impar y α(n) ≥ 3, tenemos que (ver (c) de la Observaci´on 3.12 y los comentarios siguientes al Lema 3.13) i = l(n) < n − 3 es impar, que 1 n+i+1 λi ≡ (5) (2) 2 n y que i n+2 (6) λi ≡ λi (2).

2 2

80

Enrique Torres Giese

Por otro lado n ≡ 1, 3 (4). Si n ≡ 1 (4), entonces l(n) ≡ 1 (4), ν 2i ≥ 1 = 0. Utilizando la ecuaci´on (6) tenemos que λi ≡ 0 (2) y de y ν n+2 2 acuerdo a (5) 1 n+i+1 0≡ (2) n 2 lo cual contradice la definici´on de l(n). Si n ≡ 3 (4) , entonces l(n) ≡ 3 (4), ν 2i = 0 y ν n+2 ≥ 1. Utilizando (6) tenemos que λi ≡ 0 (2) lo 2 cual, de acuerdo a (5) contradice la definici´on de l(n). Esto concluye la demostraci´on del primer inciso.

Si n = 2s + 1 con s ≥ 1 y L2n+1 (2m ) ⊆ R2n+1+2(n−2)−1 se tiene el siguiente diagrama B(n + 1, 2(n − 2) − 1)

BSO(2(n − 2) − 1)

p2n−5 f¯2(n−2)−1

B(n + 1, 2(n − 2))

f¯2(n−2)

L2n+1 (2m )

g

BU (n + 1)

BSO(2n + 2)

La ecuaci´ on (2) toma ahora la forma (ver inciso (b) de la Observaci´on 3.12) λn−2 ≡ donde ν

2n−1 n

1 n+i+1 1 2n − 1 ≡ 2 2 n n

(2)

= 1, por lo que λn−2 ≡ 1 (2), y adem´as ∗ λn−2 z n−2 = f¯2(n−2) (an−2 ) ∗ ¯ = f2(n−2)−1 (p∗2n−5 (an−2 )) = f¯∗ (bn−2 ) 2(n−2)−1

=

1 2n − 2 n−2 z 2 n

donde ν 2n−2 = 2 + α(n − 2) − 1 ≥ 2, por lo que z n−2 = 0, que es una n contradicci´ on. Con ´esto concluye la demostraci´on del Teorema 3.10.

81

No-inmersi´on de espacios lente

Observaci´ on 3.14 En la demostraci´on del Teorema 3.10 s´olo se utilizaron relaciones obtenidas a partir de Sq 2 y Sq 4 mientras que en el Teorema 3.8 se obtuvo Sq 2k (aj ) para k ≥ 1. Surge la pregunta natural: qu´e ocurre con las dem´as relaciones que proporcionan Sq 2k para k ≥ 3.

Utilizando la misma notaci´ on que en el Teorema anterior y de acuerdo al Teorema 3.8 se tiene que para k + i ≤ n i λi ≡ k

i 1 n+i+k i−1 1 n+i+k−1 + k 2 n k−1 2 n +

i−k+2 1 n+i+2 2 2 n

+(i − k + 1)

1 n+i+1 2 n

n+1 +... n

n+k−2 n

n+k−1 n+k + λi n n

(2).

Lo cual, de acuerdo a la definici´on de l(n), se convierte en la relaci´on (7)

1 n+i+1 i λi ≡ (i − k + 1) 2 k n

n+k−1 n+k + λi n n

(2).

Cuando n + 1 ≤ k + i la relaci´on que obtenemos es (8)

0 ≡ (i − k + 1)

1 n+i+1 2 n

n+k−1 n+k + λi n n

(2)

pues Sq 2k (ai ) = Sq 2k (z i ) = 0. Cuando n es par y α(n) ≥ 2 vimos en la demostracion del Teorema que i = l(n) es par ≤ n − 2 y que λi ≡ 0, luego las ecuaciones (7) y (8) se transforman en (9)

0 ≡ (k + 1)

1 n+i+1 n 2

n+k−1 n

la cual s´olo tiene inter´es de ser estudiada para k par, adem´as de que k est´a restringido por la relaci´on k ≤ i = l(n). Recu´erdese que 2i es la codimensi´on de la inmersi´on supuesta. De modo que la ecuaci´on (9) no conduce a mejora alguna de nuestro resultado. Similarmente, para n impar y α(n) ≥ 3 los t´erminos que involucran a i = l(n) en las ecuaciones (7) y (8) no conducen a mejora alguna.

Cuando n = 2s + 1 se tiene que i = l(n) = n − 2 y la ecuaci´on (7) es v´alida para k ≤ 2. El caso k = 1 fue estudiado en la prueba, mientras

82

Enrique Torres Giese

que en el caso k = 2 la ecuacion (7) es la ecuaci´on (2.4) que se reduce a la relaci´on 2i ≡ n+2 que no tiene mayor importacia. Por otro lado, la 2 ecuaci´on (8) toma la forma 0 ≡ k n+k−1 pues 12 2n−1 ≡ 1, y ´esta s´olo n n ≥ 1. tendr´ıa inter´es para k impar, pero en tal caso ν n+k−1 n Con este argumento queda eliminada la posibilidad de mejorar nuestro resultado con las relaciones que producen Sq 2k (ai ) con k ≥ 3.

Agradecimientos El autor agradece al Dr. Jes´ us Gonz´alez su apoyo y valiosa direcci´on durante la elaboraci´on de este trabajo.

Enrique Torres Giese Mathematics Department, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI 53706, torres@math.wisc.edu

Referencias [1] Davis D. M.; Mahowald M. E., Immersions of complex projective spaces and the generalized vector field problem. Proc. London Math. Soc., 35 (1977), 333–344.

[2] Dold A., Lectures on Algebraic Topology. Springer-Verlag, BerlinNew York, 1980.

[3] Gonz´alez J., Connective K-theoretic Euler classes and nonimmersions of 2k lens spaces. J. London Math. Soc., 63 (2001), 247-256.

[4] Gonz´alez J., Inmersi´on de variedades: Una excursi´on homot´opica, Aportaciones Matem´aticas. Comunicaciones 30. Memorias del XXXIV Congreso Nacional de la Sociedad Matem´atica Mexicana, (2002), 131-164.

[5] Hatcher A., Algebraic Topology. Disponible a trav´es de la red en: http://www.math.cornell.edu/˜hatcher.

[6] Hatcher A., Vector Bundles and K-theory. Disponible a trav´es de la red en: http://www.math.cornell.edu/˜hatcher.

[7] Hirsch M. W., Immersions of manifolds. Trans. Amer. Math.

Soc., 93 (1959), 242–276.

No-inmersi´on de espacios lente

83

[8] Husemoller D., Fibre Bundles. Graduate Texts in Mathematics, Springer–Verlag, 1993.

[9] Junod B., A non-immersion result for Lens Spaces Ln (2m ). Math.

J. Okayama Univ., 37 (1995), 137–151.

[10] McCleary J., A User’s Guide to Spectral Sequences. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2000.

[11] Milgram R. J., Immersing projective spaces. Ann. of Math, 85 (1967), 473-482.

[12] Mosher R. E.; Tangora M. C., Cohomology Operations and Applications in Homotopy Theory. Harper & Row, New York-London, 1968.

[13] Sanderson B. J.; Schwarzenberger R. L. E., Non-immersion theorems for differentiable manifolds. Proc. Cambridge Philos. Soc., 59 (1963), 222–319.

[14] Spanier E., Algebraic Topology. Mc Graw–Hill. New York, 1966.

[15] Switzer R., Algebraic Topology–Homotopy and Homology.

Springer-Verlag, New York, 1975.

Morfismos, Comunicaciones Estudiantiles del Departamento de Matem´aticas del CINVESTAV, se termin´o de imprimir en el mes de noviembre de 2004 en el taller de reproducci´ on del mismo departamento localizado en Av. IPN 2508, Col. San Pedro Zacatenco, M´exico, D.F. 07300. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

Contenido Multiobjective Markov Control Processes: a Linear Programming Approach On´esimo Hern´ andez-Lerma and Rosario Romera . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Tutte uniqueness of locally grid graphs D. Garijo, A. M´ arquez and M.P. Revuelta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

No-inmersi´on de espacios lente Enrique Torres Giese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57