Morfismos, Vol 22, No 1, 2018

VOLUMEN 22 NÚMERO 1 ENERO A JUNIO DE 2018 ISSN: 1870-6525

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VOLUMEN 22 NÚMERO 1 ENERO A JUNIO DE 2018 ISSN: 1870-6525

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Morfismos

Contents - Contenido Hochschild homology and cohomology for involutive A∞ -algebras Ramsès Fernàndez-València . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Baker-Gross theorem revisited José Juan-Zacarı́as . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Non-contractible configuration spaces Cesar A. Ipanaque Zapata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Morfismos, Vol. 22, No. 1, 2018, pp. 1–15 Morfismos, Vol. 22, No. 1, 2018, pp. 1–15

Hochschild homology and cohomology for involutive A∞-algebras Hochschild homology and cohomology for 1 involutive A∞-algebras Ramsès Fernàndez-València

Ramsès Fernàndez-València

1

Abstract We present a study of the homological algebra of bimodules over involution. Furthermore we introA∞ -algebras endowed with an Abstract duce a derived description of Hochschild cohomol-over We present a study of the homologicalhomology algebra ofand bimodules ogyAfor -algebras involutiveendowed A∞ -algebras. with an involution. Furthermore we intro∞

duce a derived description of Hochschild homology and cohomol2010 Mathematics SubjectAClassification: 16E40. ogy for involutive ∞ -algebras.

Keywords and phrases: Hochshcild homology, A∞ -algebra, resolution.

2010 Mathematics Subject Classification: 16E40.

Keywords and phrases: Hochshcild homology, A∞ -algebra, resolution.

1

Introduction

Hochschild homology and cohomology are homology and cohomology 1 Introduction theories developed for associative algebras which appears naturally when one studies its deformation theory. Furthermore, hoHochschild homology and cohomology are homologyHochschild and cohomology mology playsdeveloped a central role topologicalalgebras field theory in order to describe theories for inassociative which appears naturally thewhen closedone states partits of deformation a topologicaltheory. field theory. studies Furthermore, Hochschild hoAn involutive of Hochschild homology and cohomology was mology plays a version central role in topological field theory in order to describe developed by states Braun part in [1]ofby consideringfield associative the closed a topological theory. and A∞ -algebras endowed an involution and morphisms which commute with thewas Anwith involutive version of Hochschild homology and cohomology involution. developed by Braun in [1] by considering associative and A∞ -algebras This paper takes step further with regards towhich [5]. Whilst in the lat-the endowed with an ainvolution and morphisms commute with ter involution. paper we develop the homological algebra required to give a derived versionThis of Braun’s involutive and forlatpaper takes a stepHochschild further withhomology regards to [5].cohomology Whilst in the ter paper we develop the homological algebra required to give a derived 1 The author was supported by EPSRC grant EP/I003908/2.

2version of Braun’s involutive Hochschild homology and cohomology for This paper was part of a Ph.D. thesis submitted in August 2015 to Swansea University (Wales, United Kingdom) under the supervision of Dr. Jeffrey H. Gian1 The author was supported by EPSRC grant EP/I003908/2. siracusa.

2 This paper was part of a Ph.D. thesis submitted in August 2015 to Swansea University (Wales, United Kingdom) under the supervision of Dr. Jeffrey H. Giansiracusa.

1

1

2

Ramsès Fernàndez-València

involutive associative algebras, this research is devoted to develop the machinery required to give a derived description of involutive Hochschild homology and cohomology for A∞ -algebras endowed with an involution.

As in [5], this research has been driven by the author’s research on Costello’s classification of topological conformal field theories [2], where he proves that an open 2-dimensional theory is equivalent to a Calabi-Yau A∞ -category. In [4], the author extends the picture to unoriented topological conformal field theories, where open theories now correspond to involutive Calabi-Yau A∞ -categories, and the closed state space of the universal open-closed extension turns out to be the involutive Hochschild chain complex of the open state algebra.

2

Basic concepts

2.1

Coalgebras and bicomodules

An involutive graded coalgebra over a field K is a graded K-vector space C endowed with a coproduct ∆ : C → C ⊗K C of degree zero together with a counit ε : C → K and an involution : C → C such that: 1. The graded K-vector space C is coassociative and counital, and 2. the involution and ∆ are compatible, therefore: ∆(c ) = (∆(c)) , for c ∈ C, where the involution on C ⊗K C is given by the following expression: (c1 ⊗ c2 ) = (−1)|c1 ||c2 | c 2 ⊗ c 1 , for c1 , c2 ∈ C. An involutive coderivation on an involutive coalgebra C is a map L : C → C preserving involutions and making the following diagram commutative: C

L

C

∆

∆

C ⊗K C

L⊗IdC +IdC ⊗L

C ⊗K C

Denote with iCoder(−) the spaces of coderivations of involutive coalgebras. Observe that iCoder(−) are Lie subalgebras over Coder(−) whose Lie bracket is given by the commutator [n, −].

An involutive differential graded coalgebra is an involutive coalgebra C equipped with an involutive coderivation b : C → C of degree −1 such that b2 = b ◦ b = 0.

Hochschild homology and cohomology for involutive A∞ -algebras

3

A morphism between two involutive coalgebras C and D is a graded f

map C −→ D compatible with the involutions which makes the following diagram commutative: (1)

f

C

D

∆C

∆D

C ⊗K C

D ⊗K D

f ⊗f

Example 2.1.1. Let us suppose that A is an associative K-algebra endowed with an involution. An involutive K-bimodule M is a Kbimodule M equipped with an involution satisfying the following condition (a · m) = m · a .

For an involutive graded K-bimodule A, we define the cotensor coalgebra of A as

A⊗ K n .

TA = n≥0

We define an involution in

A⊗K n

(a1 ⊗ · · · ⊗ an ) := (−1)

by stating:

n

i=1

n

|ai |(

j=i+1

|aj |)

(a n ⊗ · · · ⊗ a 1 ).

We can endow T A with a coproduct as follows: ∆(a1 ⊗ · · · ⊗ an ) =

n

i=0

(a1 ⊗ · · · ⊗ ai ) ⊗ (ai+1 ⊗ · · · ⊗ an ).

Observe that ∆ commutes with the involution.

For a given (involutive) graded algebra A, HomA−iBimod (−, −) and iCoder(−) will denote the spaces of involutive homomorphisms and coderivations of involutive A-bimodules respectively. We will write HomA−Bimod (−, −) for the space of homomorphisms of A-bimodules.

Let us think of A as a (involutive) bimodule over itself. We denote the suspension of A by SA and define it as the graded (involutive) Kbimodule with SAi = Ai−1 . Given such a bimodule A, we define the following morphism of degree −1 induced by the identity s : A → SA by s(a) = a.

Proposition 2.1. Let us define Bar(A) := A ⊗K T SA ⊗K A. For an involutive graded algebra A, the following canonical isomorphism of complexes holds: iCoder (T SA) ∼ = HomA−iBimod (Bar(A), A) ,

4

Ramsès Fernàndez-València

where the involution we endow Bar(A) with is the following: (a0 ⊗ · · · ⊗ an+1 ) = a n+1 ⊗ · · · ⊗ a 0 .

Proof. The proof follows the arguments in Proposition 4.1.1 [5], where we show the result for the non-involutive setting in order to restrict to the involutive one.

The degree −n part of HomA−Bimod (Bar(A), A) is the space of degree −n linear maps T SA → A, which is isomorphic to the space of degree (−n − 1) linear maps T SA → SA. By the universal property of the tensor coalgebra, there is a bijection between degree (−n − 1) linear maps T SA → SA and degree (−n − 1) coderivations on T SA. Hence the degree n part of HomA−Bimod (Bar(A), A) is isomorphic to the degree n part of Coder (T SA). One checks directly that this isomorphism restricts to an isomorphism of graded vector spaces HomA−iBimod (Bar(A), A) ∼ = iCoder (T SA) .

Finally, one can check that the differentials coincide under the above isomorphism, cf. Section 12.2.4 [7].

Remark 2.2. Proposition 2.1 allows us to think of a coderivation on the coalgebra T SA as a map T A → A. Such a map f : T A → A can be described as a collection of maps {fn : A⊗n → A} which will be called the components of f . b2

If b is a coderivation of degree −1 on T A with bn : A⊗K n → A, then becomes a linear map of degree −2 with b2n

=

n−1

i+j=n+1 k=0

bi ◦ Id⊗k ◦ bj ◦ Id⊗(n−k−j) .

The coderivation b will be a differential for T A if, and only if, all the components b2n vanish.

Lemma 2.1.2 (cf. Lemma 1.3 [6]). If bk : (SA)⊗K k → SA is an involutive linear map of degree −1, we define mk : A⊗K k → A as mk = s−1 ◦ bk ◦ s⊗K k . Under these conditions: bk (sa1 ⊗ · · · ⊗ sak ) = σmk (a1 ⊗ · · · ⊗ ak ), where σ := (−1)(k−1)|a1 |+(k−2)|a2 |+···+2|ak−2 |+|ak−1 |+

k(k−1) 2

.

Hochschild homology and cohomology for involutive A∞ -algebras

5

Proof. The proof follows the arguments of Lemma 1.3 [6]. We only need to observe that the involutions are preserved as all the maps involved in the proof are assumed to be involutive.

Let mk := Ďƒmk , then we have bk (sa1 ⊗¡ ¡ ¡⊗sak ) = mk (a1 ⊗¡ ¡ ¡⊗ak ).

Proposition 2.3. Given an involutive graded K-bimodule A, let i = |a1 | + ¡ ¡ ¡ + |ai | − i for ai ∈ A and 1 ≤ i ≤ n. A boundary map b on T SA is given in terms of the maps mk by the following formula: bn (sa1 ⊗ ¡ ¡ ¡ ⊗ san ) =

n n−k+1

k=0

i=1

(−1) i−1 (sa1 ⊗ ¡ ¡ ¡

¡ ¡ ¡ ⊗ sai−1 ⊗ mk (ai ⊗ ¡ ¡ ¡ ⊗ ai+k−1 ) ⊗ ¡ ¡ ¡ ⊗ san ).

Proof. This proof follows the arguments of Proposition 1.4 [6]. The only detail that must be checked is that bn preserves involutions: bn ((sa1 ⊗ ¡ ¡ ¡ ⊗ san ) )

Âą(sa n ⊗ ¡ ¡ ¡ ⊗ sa j ⊗ mk (a j−1 ⊗ ¡ ¡ ¡ ⊗ a j−k+1 ) ⊗ ¡ ¡ ¡ ⊗ sa 1 ) = j,k

=

j,k

Âą(sa1 ⊗ ¡ ¡ ¡ ⊗ mk (aj−k+1 ⊗ ¡ ¡ ¡ ⊗ aj−1 ) ⊗ saj ⊗ ¡ ¡ ¡ ⊗ san )

= (bn (sa1 ⊗ ¡ ¡ ¡ ⊗ san )) .

Given an involutive coalgebra C with coproduct ∆C and counit Îľ, for an involutive graded vector space P , a left coaction is a linear map ∆L : P → C ⊗K P such that 1. (Id ⊗ ∆C ) â—Ś ∆L = (∆C ⊗ Id) â—Ś ∆L ; 2. (Id ⊗ Îľ) â—Ś ∆L = Id. Analagously we introduce the concept of right coaction.

Given an involutive coalgebra (C, ∆C , Îľ) with involution we define an involutive C-bicomodule as an involutive graded vector space P with involution †, a left coaction ∆L : P → C ⊗K P and a right coaction ∆R : P → P ⊗K C which are compatible with the involutions, that is the diagrams below commute:

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Ramsès Fernàndez-València

(2)

(−)

P

P

∆L

∆R

C ⊗K P (3)

(−,−) ∆L

P

P ⊗K C C ⊗P

∆R

P ⊗K C Here

∆L ⊗IdC

(−, −) : C ⊗K P c⊗p

IdC ⊗∆R

C ⊗ K P ⊗K C

→ P ⊗K C → p† ⊗ c .

For two involutive C-bicomodules (P1 , ∆1 ) and (P2 , ∆2 ), a morphism f

P1 −→ P2 is defined as an involutive morphism making diagrams below commute: (4)

P1

∆L 1

C ⊗K P1

f

P2

2.2

∆L 2

IdC ⊗f

C ⊗K P2

(5)

P1

∆R 1

P1 ⊗ K C

f

P2

∆R 2

IdC ⊗f

P2 ⊗ K C

A∞ -algebras and A∞ -quasi-isomorphisms

An involutive A∞ -algebra is an involutive graded vector space A endowed with involutive morphisms bn : (SA)⊗K n → SA, n ≥ 1,

(6)

of degree n − 2 such that the identity below holds:

(−1)i+jl bi+1+l ◦ (Id⊗i ⊗ bj ⊗ Id⊗l ) = 0, ∀n ≥ 1.

(7) i+j+l=n

An (involutive) A∞ -algebra A is called strictly unital if there exists an element 1A ∈ A0 which is a unit for b2 , satisfying the following conditions bn (a1 , . . . , 1, . . . , an ) = 0 if n = 2 and 1 A = 1A . If the map b0 : K → SA is non trivial, then we say that A is a curved A∞ -algebra, it will be called flat otherwise.

Hochschild homology and cohomology for involutive A∞ -algebras

7

Remark 2.4.

1. It is a straight computation to check that condition (7) says, in particular, that b21 = 0.

2. Observe that when one applies (7) we need to take care of signs due to Koszul sign rule: (f ⊗ g)(x ⊗ y) = (−1)|g||x| f (x) ⊗ g(x).

Example 2.2.1.

1. The concept A∞ -algebra is a generalization for that of a differential graded algebra. Indeed, if the maps bn = 0 for n ≥ 3 then A is a differential Z-graded algebra and conversely an A∞ -algebra A yields a differential graded algebra if we require bn = 0 for n ≥ 3.

2. The definition of A∞ -algebra was introduced by Stasheff whose motivation was the study of the graded abelian group of singular chains on the based loop space of a topological space.

For an involutive A∞ -algebra (A, bA ), the involutive bar complex is the involutive differential graded coalgebra T SA, endowed with a coderivation defined by bi = s−1 ◦ mi ◦ s⊗K i (cf. Definition 1.2.2.3 [9]).

Given two involutive A∞ -algebras C and D, a morphism of involutive A∞ -algebras f : C → D is an involutive morphism of degree zero between the associated involutive differential graded coalgebras T SC → T SD. It follows from Proposition 2.1 that the definition of an involutive A∞ -algebra can be summarized by saying that it is an involutive graded K-vector space A equipped with an involutive coderivation on Bar(A) of degree −1.

Remark 2.5. It follows from [1], Definition 2.8, we have that a morphism of involutive A∞ -algebras f : C → D can be given by a series of involutive homogeneous maps of degree zero fn : (SC)⊗K n → SD, n ≥ 1, such that

⊗l fi+l+1 ◦ Id⊗i ⊗ b ⊗ Id (8) j SC SC = i+j+l=n

i1 +···+is =n

bs ◦(fi1 ⊗· · ·⊗fis ).

The composition f ◦ g of two morphisms of involutive A∞ -algebras is given by

fs ◦ (gi1 ⊗ · · · ⊗ gis ); (f ◦ g)n = i1 +···+is =n

8

Ramsès Fernàndez-València

the identity on SC is defined as f1 = IdSC and fn = 0 for n ≼ 2.

The condition of being involutive means that the following identity holds: fn (c1 , . . . , cn ) = Ďƒfn (c n , . . . , c 1 ), where Ďƒ := (−1)

n

i=1

n

|ci |(

j=i+1

|cj |)

(−1)

n(n+1) −1 2

(see [1], Definition 2.7).

For an involutive A∞ -algebra A, we define its associated homology algebra H• (A) as the homology of the differential b1 on A, that is: H• (A) = H• (A, b1 ).

Remark 2.6. Endowed with b2 as multiplication, the homology of an A∞ -algebra A is an associative graded algebra, whereas A is not usually associative.

Let f : A1 → A2 be a morphism of involutive A∞ -algebras with components fn ; we note that for n = 1, f1 induces a morphism of algebras H• (A1 ) → H• (A2 ). We say that f : A1 → A2 is an A∞ -quasiisomorphism if f1 is a quasi-isomorphism.

2.3

A∞ -bimodules

Let (A, bA ) be an involutive A∞ -algebra. An involutive A∞ -bimodule is a pair (M, bM ) where M is a graded involutive K-vector space and bM is an involutive differential on the Bar(A)-bicomodule, whose involution will be introduced shortly: B(M ) := Bar(A) ⊗K SM ⊗K Bar(A).

If denotes the involution of Bar(A) and †is the involution for M , we can endow B(M ) with the following involution: (a1 , . . . , an , m, a 1 , . . . , a n )‥ := ((a 1 , . . . , a n ) , m†, (a1 , . . . , an ) ).

Let M, bM and N, bN be two involutive A∞ -bimodules. We define a morphism of involutive A∞ -bimodules f : M → N as a morphism of Bar(A)-bicomodules F : B(M ) → B(N ) such that bN â—Ś F = F â—Ś bM .

Proposition 2.7 (cf. [6] Proposition 3.4). If f : A1 → A2 is a morphism of involutive A∞ -algebras, then A2 becomes an involutive bimodule over A1 .

Hochschild homology and cohomology for involutive A∞ -algebras

9

Remark 2.8 (Section 5.1 [8]). Let iVect be the category of involutive Z-graded vector spaces and involutive morphisms. For an involutive A∞ -algebra A, involutive A-bimodules and their respective morphisms form a differential graded category. Indeed, following [8], Definition 5.1.5: let A be an involutive A∞ -algebra and let us define the category A − iBimod whose class of objects are involutive A-bimodules and where HomA−iBimod (M, N ) is: HomniVect (Bar(A) ⊗K SM ⊗K Bar(A), Bar(A) ⊗K SN ⊗K Bar(A)).

Let us recall that HomniVect (Bar(A) ⊗K SM ⊗K Bar(A), Bar(A) ⊗K SN ⊗K Bar(A)) is by definition the product over i ∈ Z of the morphism sets HomiVect ((Bar(A)⊗K SM ⊗K Bar(A))i , (Bar(A)⊗K SN ⊗K Bar(A))i+n ).

The morphism HomniVect (Bar(A) ⊗K SM ⊗K Bar(A), Bar(A) ⊗K SN ⊗K Bar(A)) → Homn+1 iVect (Bar(A) ⊗K SM ⊗K Bar(A), Bar(A) ⊗K SN ⊗K Bar(A))

sends a family {fi }i∈Z to a family {bN ◦fi −(−1)n fi+1 ◦bM }i∈Z . Observe that the zero cycles in Hom•iVect (Bar(A) ⊗K M ⊗K Bar(A), Bar(A) ⊗K N ⊗K Bar(A)) are precisely the morphisms of involutive A-bimodules.

This morphism defines a differential, indeed: for fixed indices i, n ∈ Z we have

d2 (fi ) = d bN fi − (−1)n fi+1 bM

= bN bN fi − (−1)n fi+1 bM − (−1)n+1 bN fi − (−1)n fi+1 bM bM (!)

= −(−1)n bN fi+1 bM − (−1)n+1 bN fi+1 bM = 0,

where (!) points out the fact that bN ◦ bN = 0 = bM ◦ bM .

For a morphism φ ∈ HomniVect (Bar(A) ⊗K M ⊗K Bar(A), Bar(A) ⊗K N ⊗K Bar(A)) and an element x ∈ Bar(A)⊗K M ⊗K Bar(A), the complex HomA−iBimod (M, N ) becomes an involutive complex if we endowed it with the involution φ (x) = φ(x ).

The functor HomA−iBimod (M, −) sends an involutive A-bimodule F to the involutive K-vector space HomA−iBimod (M, F ) of involutive homomorphisms. Given a homomorphism f : F → G, for

F, G ∈ Obj A − iBimod ,

10

Ramsès Fernàndez-València

HomA−iBimod (M, −) sends f to the involutive map: f : HomA−iBimod (M, F ) → HomA−iBimod (M, G) . φ → f ◦φ We prove that f preserves involutions: (f φ )(x) = (f ◦ φ )(x) = f (φ(x ))

= f ((φ(x)) ) = (f (φ(x))) = (f φ(x)) .

Let us introduce the functor HomA−iBimod (−, M ), which sends an

involutive homomorphism f : F → G, for F, G ∈ Obj A − iBimod , to ϕ : HomA−iBimod (G, M ) → HomA−iBimod (F, M ) φ → φ◦f Let us check that the involution is preserved: ϕ(φ )(x) = (φ ◦f )(x) = φ(f (x) ) = φ(f (x )) = ϕ(φ)(x ) = (ϕ(φ)) (x).

Let A be an involutive A∞ -algebra and let M, bM and N, bN be involutive A-bimodules. For f, g : M → N involutive morphisms of Abimodules, an A∞ -homotopy between f and g is an involutive morphism h : M → N of A-bimodules satisfying f − g = bN ◦ h + h ◦ bM .

We say that two morphisms u : M → N and v : N → M of involutive A-bimodules are homotopy equivalent if u ◦ v ∼ IdN and v ◦ u ∼ IdM .

3

The involutive tensor product

For an involutive A∞ -algebra A and involutive A-bimodules M and N , ∞ N is the following object in iVectK : the involutive tensor product M ∞ N := M

M ⊗K Bar(A) ⊗K N . (m ⊗ a1 ⊗ · · · ⊗ ak ⊗ n − m ⊗ a1 ⊗ · · · ⊗ ak ⊗ n )

∞ N of the form m⊗a1 ⊗· · ·⊗ak ⊗n, Observe that, for an element of M we have: (m ⊗ a1 ⊗ · · · ⊗ ak ⊗ n) = m ⊗ a1 ⊗ · · · ⊗ ak ⊗ n = m ⊗ a1 ⊗ · · · ⊗ ak ⊗ n .

Hochschild homology and cohomology for involutive A∞ -algebras

11

Proposition 3.1. For an involutive A and involutive A A∞ -algebra

bimodules M, N and L, HomiVect M ∞ N, L is isomorphic to

M ⊗K Bar(A) , HomA−iBimod (N, L) , HomiVect âˆź

where in M ⊗K Bar(A) : (m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ) = m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak , âˆź denotes the relation

and

M ⊗K Bar(A) âˆź

m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak = m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak has the identity map as involution.

∞ N → L be an involutive map. We define: Proof. Let f : M

M ⊗K Bar(A) Ď„ (f ) := Ď„f ∈ HomiVect , HomA−iBimod (N, L) , âˆź where Ď„f (m⊗a1 ⊗¡ ¡ ¡⊗ak ) := Ď„f [m⊗a1 ⊗¡ ¡ ¡⊗ak ] ∈ HomA−iBimod (N, L).

Finally, for n ∈ N we define: Ď„f [m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ](n) := f (m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ⊗ n) .

We need to check that τ preserves the involutions, indeed: τf [m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ](n) = f (m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ⊗ n) =

= (f (m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ⊗ n)) = (τf ) [m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ](n).

In order to see that Ď„ is an isomorphism, we build an inverse. Let us consider an involutive map g1 : m

M ⊗K Bar(A) âˆź ⊗ a1 ⊗ ¡ ¡ ¡ ⊗

→ HomA−iBimod (N, L) ak → g1 [m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ]

and define a map g2 :

∞N M → L m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ⊗ n → g1 [m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ](n)

We check that g2 is involutive:

g2 ((m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ⊗ n) ) = g2 (m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ⊗ n) =

= g1 [m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ](n) = (g1 [m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ]) (n)

= (g1 [m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ](n)) = (g2 (m ⊗ a1 ⊗ ¡ ¡ ¡ ⊗ ak ⊗ n)) .

The rest of the proof is standard and follows the steps of Theorem 2.75 [10] or Proposition 2.6.3 [11].

12

tor

Ramsès Fernàndez-València

∞ M as the covariant funcFor an A-bimodule M , let us define (−) ∞M (−)

A − iBimod −−−−−−→ A − iBimod . ∞M B

B

f ∞ IdM f ∞M . ∞M − This functor sends a map B1 −→ B2 to B1 −−−−−→ B2 ∞ M is involutive: let us consider an involutive The functor (−) map f : B1 → B2 and its image under the tensor product functor, ∞ IdM . Hence: g = f

g((b, a) ) = g(b , a) = (f (b ), a) = (f (b), a) = (g(b, a)) .

Given an involutive A∞ -algebra A, we say that an involutive Abimodule F is flat if the tensor product functor ∞ F : A − iBimod → A − iBimod (−)

is exact, that is: it takes quasi-isomorphisms to quasi-isomorphisms.

Lemma 3.0.1. If P and Q are homotopy equivalent as involutive A∞ -bimodules then, for every involutive A∞ -bimodule M , the following quasi-isomorphism in the category of involutive A∞ -bimodules holds: ∞ M Q ∞ M. P

Proof. Let f : P Q : g be a homotopy equivalence. It is clear that

Therefore, we have:

and

∞ IdM ∼ k ∞ IdM . h ∼ k ⇒ h

∞M P

p a

∞M Q

q a

∞M → Q

→ f (p) a

∞M → P

→ g(q) a

∞M → P

→ g(f (p)) a ∞M → Q

→ f (g(q)) a

the result follows since f ◦ g ∼ IdQ and g ◦ f ∼ IdP .

Lemma 3.0.2. Let A be an involutive A∞ -algebra. If P and Q are homotopy equivalent as involutive A-bimodules then, for every involutive A-bimodule M , the following quasi-isomorphism holds: HomA−iBimod (P, M ) HomA−iBimod (Q, M ).

Hochschild homology and cohomology for involutive A∞ -algebras

13

Proof. Consider f : P → Q a homotopy equivalence and let g : Q → P be its homotopy inverse. If [−, −] denotes the homotopy classes of morphisms, then both f and g induce the following maps: f : [P, M ] → [Q, M ] α → α ◦ g g : [Q, M ] → [P, M ] β → β ◦ f Now we have: f ◦ g ◦ β = f ◦ β ◦ f = β ◦ g ◦ f ∼ β; g ◦ f ◦ α = g ◦ α ◦ g = α ◦ f ◦ g ∼ α.

4 4.1

Involutive Hochschild (co)homology Hochschild homology for involutive A∞ -algebras

We define the involutive Hochschild chain complex of an involutive A∞ algebra A with coefficients in a involutive A-bimodule M as ∞ B(A).

C•inv (M, A) = M

The differential is the same given in Section 7.2.4 [8]. The involutive Hochschild homology of A with coefficients in M is HHn (M, A) = HCninv (M, A).

Lemma 4.1.1. For an involutive flat strictly unital A∞ -algebra A and an involutive A-bimodule M , the following quasi-isomorphism holds: ∞ A. C•inv (M, A) M

Proof. The result follows from:

∞A M ∞ B(A) = C•inv (M, A).

M

Observe that we are using that there is a quasi-isomorphism, therefore a homotopy equivalence (Proposition 1.3.5.1 [9]), between B(A) and A (Proposition 2, Section 2.3.1 [3]).

14

4.2

Ramsès Fernàndez-València

Hochschild cohomology for involutive A∞ -algebras

The involutive Hochschild cochain complex of an involutive A∞ -algebra A with coefficients on an involutive A-bimodule M is defined as the • K-vector space Cinv (A, M ) := HomA−iBimod (B(A), M ), with the differential defined in section 7.1 of [8].

Proposition 4.1. For an involutive A∞ -algebra A and an involutive A• (A, M ) bimodule M , we have the following quasi-isomorphism: Cinv HomA−iBimod (A, M ).

Proof. The result follows from: • Cinv (A, M ) = HomA−iBimod (B(A), M ) := (!)

HomniVect (Bar(A) ⊗K SBar(A) ⊗K Bar(A), Bar(A) ⊗K SM ⊗K Bar(A)) HomniVect (Bar(A) ⊗K SA ⊗K Bar(A), Bar(A) ⊗K SM ⊗K Bar(A)) =: HomA−iBimod (A, M ).

Here (!) points out the fact that SBar(A) is a projective resolution of SA in iVect and hence we have the quasi-isomorphism SBar(A) SA. Observe that SBar(A) is projective in iVect, therefore the involved functors in the proof are exact and preserve quasi-isomorphisms.

Acknowledgement The author thanks and highly appreciates the comments and suggestions made by the anonimous reviewer, which significantly contributed to improving the quality of the publication.

Ramsès Fernàndez-València Technology Centre of Catalonia, 54 Antic de València Road, 08005 Barcelona, Catalonia ramses.fernandez.valencia@gmail.com

References [1] Braun C., Involutive A∞ -algebras and dihedral cohomology, Journal of Homotopy and Related Structures, 9 (2014), 317-337.

[2] Costello K.J., Topological conformal field theories and Calabi-Yau categories, Adv. Math., 210 (2007), 165–214.

[3] Ferrario A., A∞ -bimodules in deformation quantization, Ph.D. thesis, Eidgenössische Technische Hochschule Zürich (2012).

Hochschild homology and cohomology for involutive A∞ -algebras

15

[4] Fernàndez-València R., On the structure of unoriented topological conformal field theories, Geometriae Dedicata Volume 189 Issue 1 (2017), 113–138. https://doi.org/10.1007/s10711-017-0220-6 [5] Fernàndez-València R., Giansiracusa J., On the Hochschild homology of involutive algebras, Glasgow Mathematical Journal 60, Issue 1 (2018), 187–198. https://doi.org/10.1017/S0017089516000653 [6] Getzler E., Jones J., A∞ -algebras and the cyclic Bar complex, Illinois J. Math.

34 (1990), 256–283 [7] Loday J.L., Vallette B., Algebraic Operads, Grundlehren der Mathematischen Wissenschaften, 346, Springer-Verlag (2012).

[8] Kontsevich M., Soibelman Y., Notes on A∞ -algebras, A∞ -categories and noncommutative geometry, Homological Mirror Symmetry, Lecture Notes in Phys., 757, Springer (2009).

[9] Lefèvre-Hasegawa K., Sur les A∞ -catégories, Ph.D. thesis, Université Paris 7 - Denis Diderot, 2003 [10] Rotman J., An Introduction to Homological Algebra, 2n ed., Springer-Verlag (2009).

[11] Weibel C., An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, 354, Cambridge University Press (1994).

Morfismos, Vol. 22, No. 1, 2018, pp. 17–25 Morfismos, Vol. 22, No. 1, 2018, pp. 17–25

Baker-Gross theorem revisited Baker-Gross theorem revisited José Juan-Zacarı́as José Juan-Zacarı́as Abstract F. Gross conjectured that any meromorphic solution of the Fermat Cubic F3 : x3 + y 3 = 1 areAbstract elliptic functions composed with entire functions. The conjecture was solved affirmatively first by F. Gross conjectured that any meromorphic solution of the FerI. N. Baker who found3 explicit formulas of those elliptic functions mat Cubic F3 : x + y 3 = 1 are elliptic functions composed with and later F. Gross gave another proof proving that in fact one of entire functions. The conjecture was solved affirmatively first by them uniformize the Fermat cubic. In this paper we give an alterI. N. Baker who found explicit formulas of those elliptic functions native proof of the Baker and Gross theorems. With our method and later F. Gross gave another proof proving that in fact one of we obtain other analogous formulas. Some remarks on Fermat them uniformize the Fermat cubic. In this paper we give an altercurves of higher degree is given. native proof of the Baker and Gross theorems. With our method we obtain other analogous formulas. Some remarks on Fermat 2010 Mathematics Subject Classification: 30D30, 30F10. curves of higher degree is given.

Keywords and phrases: Fermat curves, elliptic functions, Baker-Gross.

2010 Mathematics Subject Classification: 30D30, 30F10.

Keywords and phrases: Fermat curves, elliptic functions, Baker-Gross.

Introduction

Consider the Fermat cubic Introduction (1) Consider the Fermat cubic F3 : x3 + y 3 = 1.

3 curve, 3 This(1) algebraic curve defines anFelliptic i.e., a compact Riemann 3 : x + y = 1. surface of genus 1 (taking the zeros in CP2 of its homogenization). A meromorphic solution of defines this equation is, bycurve, definition, pair of meroThis algebraic curve an elliptic i.e., a acompact Riemann +2g 3of=its1. homogenization).

In his paper [2] A morphic functions planethe such thatinf 3CP surface of genusin1the (taking zeros F. Gross conjectures that any meromorphic solution of the Fermat meromorphic solution of this equation is, by definition, a paircubic of mero3 3 + g = 1.

In his paper [2] morphic functions in the plane such that f 1 This work was partially supported by PAPIIT IN100811. The present paper F. Gross that any meromorphic solution of the Fermat cubic contains part ofconjectures the Undergraduate Thesis of the author written under the supervision

of Dr. Alberto Verjovsky at the Cuernavaca Branch of the Institute of Mathematics 1 This work was partially supported

by PAPIIT IN100811.

Thedegree present of the National Autonomous University of Mexico (UNAM).

The bachelor waspaper contains part May of the2014 Undergraduate Thesis of the author written under the supervision obtained on 23rd at the Faculty of Sciences of UNAM. of Dr. Alberto Verjovsky at the Cuernavaca Branch of the Institute of Mathematics of the National Autonomous University of Mexico (UNAM). The bachelor degree was obtained on 23rd May 2014 at the Faculty of Sciences of UNAM. 17

17

18

José Juan-Zacarı́as

is obtained by composing elliptic functions with entire functions. The conjecture was solved affirmatively by I. N. Baker in [4]. He proved that any solution is the composition of the following elliptic functions with an entire function: (2)

1 1 f (z) = 1 − 3−1/2 ℘ (z) , g(z) = 1 + 3−1/2 ℘ (z) , 2℘(z) 2℘(z)

where ℘ is the Weierstrass elliptic function satisfying (℘ )2 = 4℘3 − 1.

In what follows we denote by Λ the lattice in C that defines this ℘. In particular these functions are solutions of the Fermat cubic but these formulas differ from the analogous that appear in [2], [3], which seem to contain an error. Later, F. Gross gave another proof in [5], proving in fact that the function f in (2) gives a uniformization of the Fermat cubic (1). In our context we formulate the previous results in the following theorem: Theorem (Baker-Gross). Let Λ and ℘ be as above. Then the map C/Λ → F3 given in affine coordinates by

1 1 −1/2 −1/2 (3) z → 1−3 1+3 ℘ (z) , ℘ (z) 2℘(z) 2℘(z)

is a biholomorphism between the two elliptic curves. Then by the lifting property of coverings, any pair of functions F and G, which are meromorphic in the plane and satisfy (1) have the form:

1 1 1 − 3−1/2 ℘ (α) , G = 1 + 3−1/2 ℘ (α) , (4) F = 2℘(α) 2℘(α)

where α is an entire function.

In this paper we give a proof of this theorem by using Riemann surface theory and by using an explicit map from a Weierstrass normal form to the Fermat cubic. Our proof could clarify the nature of the previous formulas, which are not obvious. Also, by this method, other formulas analogous to (3) and (4) are obtained (see (13) and (17)).

In Section 1 we recall some basic facts about elliptic curves and compute a Weierstrass normal form of the Fermat cubic, and the corresponding isomorphism as well. In the next section we prove the main theorem. Finally, in the last section we give some remarks on Fermat curves of higher degree.

Recently, N. Steinmetz communicated to the author another proof of the Gross conjecture in [7] (§2.3.5 pp. 56-57) by using Nevanlinna

19

Baker-Gross theorem

theory. He proved without reference to the Uniformization Theorem the following: Theorem (Steinmetz). Suppose that non-constant meromorphic functions f and g parametrize the algebraic curve F : xn + y m = 1

(n ≥ m ≥ 2)

1 with m + n1 < 1. Then (m, n) equals (4, 2) or (3, 3) or (3, 2). In any case f and g are given by √ m−1 f = E ◦ ψ and g = E ◦ ψ,

where E is an elliptic function satisfying E 2 = 1 − E 4 ,

E 3 = (1 − E 3 )2

and

E 2 = 1 − E 3 ,

respectively, and ψ is any non-constant entire function.

1 1.1

The normal form of the Fermat cubic Basic facts on elliptic curves

A complex elliptic curve X is by definition a compact Riemann surface of genus 1. The Plücker formula tells us that a non-singular projective curve of degree 3 in CP2 is a Riemann surface of genus 1 i.e., an elliptic curve. The reciprocal is also true and we will briefly discuss it. For this, we recall the uniformization theorem and the Weierstrass normal form.

The Uniformization Theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the Riemann sphere C, the complex plane C, or the open unit disk ∆. This theorem combined with the theory of covering spaces give us a classification of Riemann surfaces: every Riemann surface X is conformally equivalent to a quotient X̃/G, where X̃ is the universal holomorphic cover of X (hence isomorphic to one of the three previous Riemann surfaces) and G is a subgroup of holomorphic automorphisms of X̃ which acts on X̃ free and properly discontinuously. In particular, when the Riemann surface is of genus 1, it has the complex plane as its universal holomorphic cover, then X is conformally equivalent to C/Λ, for some lattice Λ ⊂ C. For an introduction to Riemann surfaces and a proof of the uniformization theorem see [1].

20

José Juan-Zacarı́as

The homogeneous polynomial with complex coefficients (5)

Y 2 Z − 4X 3 + g2 XZ 2 + g3 Z 3 ,

obtained by homogenization of the polynomial y 2 = 4x3 − g2 x − g3 ,

(6)

defines an non-singular curve if and only if the discriminant ∆ = g23 − 27g32 does not vanish. Hence, (5) defines an elliptic curve if and only if ∆ = 0. We call a Weierstrass normal form of an elliptic curve X an elliptic curve given by an equation of the form (5) which is isomorphic as a Riemann surface to X. Recall also that given a lattice Λ ⊂ C we can associate the Weierstrass elliptic function ℘ or ℘Λ given by the series: (7)

1 1 1 ℘(z) = 2 + . − z (z + ω)2 ω 2 ∗ ω∈Λ

This function satisfies the differential equation (8)

(℘ )2 = 4℘3 − g2 ℘ − g3 ,

where g2 and g3 are constants depending on Λ given by: g2 = 60

1 , ω4 ∗

ω∈Λ

g3 = 140

1 , ω6 ∗

ω∈Λ

satisfying ∆ = g23 − 27g32 = 0. Thus this function gives us a map Ψ : C/Λ → E, in affine coordinates given by: (9)

Ψ(z) = (℘(z), ℘ (z)),

from C/Λ to the elliptic curve E : y 2 = 4x3 − g2 x − g3 . This map is an biholomorphism which sends Λ to the point at infinity [0 : 1 : 0].

From the previous results and the Uniformization Theorem we can conclude that every elliptic curve has a Weierstrass normal form. Also, it is true that given a non-singular equation (6), there exists a lattice Λ with the same constants g2 and g3 . For more information, refer to [6, p. 176].

21

Baker-Gross theorem

1.2

Computing the Weierstrass normal form of the Fermat cubic

Although a Weierstrass normal form is in general difficult to compute starting from an abstract Riemann surface of genus 1, the case of the Fermat cubic is relatively easy by choosing suitable changes of variables.

Since this process will be applied to other Fermat curves in Section 3, we describe it step-by-step below: 1. Change (x, y) to (x − y, x + y) in order to eliminate the cubic term y 3 . Obtaining: E1 : 2x3 + 6xy 2 = 1.

2. Change (x, y) to (1/x, y/x) to get: E2 : 2 + 6y 2 = x3 .

3. At this point, we could use any change of variables for which the 2 3 coefficient √ of y is 1 and the coefficient of x is 4, for instance with (x, y/ 24) we obtain the case g2 = 0 and g3 = 8: E3 : y 2 = 4x3 − 8.

Observe that we obtain a map from the curve obtained in the change of variable to the original curve. For example in step 1 we obtain E1 → F3 , (x, y) → (x − y, x + y). Then, the maps associated to the previous changes of variables are: E 3 → E2

y (x, y) → x, √ , 24

(10)

E 2 → E1

(x, y) → x1 , xy ,

E1 → F3 (x, y) → (x − y, x + y).

The inverse maps are (in the reverse order, respectively): (11) (x, y) →

F3 → E1

y+x y−x , , 2 2

E1 → E2

(x, y) → x1 , xy ,

E 2 → E3 (x, y) → (x,

√ 24y).

So in each step we have a birrational isomorphism between these non-singular algebraic curves, hence a biholomorphism between their

22

JoseĚ Juan-ZacarÄąĚ as

Riemann surfaces. So we obtain, composing the maps of (11) and (10), respectively, the biholomorphisms ÎŚ : F3 â&#x2020;&#x2019; E3 and ÎŚâ&#x2C6;&#x2019;1 : E3 â&#x2020;&#x2019; F3 :

2 â&#x2C6;&#x161; yâ&#x2C6;&#x2019;x ÎŚ(x, y) = (12) , 24 , y+x y+x

y y 1 1 ÎŚâ&#x2C6;&#x2019;1 (x, y) = â&#x2C6;&#x2019;â&#x2C6;&#x161; , +â&#x2C6;&#x161; . x 24x x 24x

2

Proof of the Baker-Gross theorem

From the previous explicit formulas the Baker-Gross theorem follows easily. Consider Î&#x203A; associated to g2 = 0 and g3 = 8 and consider the biholomorphism Ψ : C/Î&#x203A; â&#x2020;&#x2019; E3 defined in (9), then the composition ÎŚâ&#x2C6;&#x2019;1 â&#x2014;Ś Ψ : C/Î&#x203A; â&#x2020;&#x2019; F3 is a biholomorphism,

1 â&#x201E;&#x2DC; (z) 1 1 â&#x201E;&#x2DC; (z) 1 â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;â&#x2C6;&#x161; , +â&#x2C6;&#x161; .

(13) ÎŚ â&#x2014;Ś Ψ(z) = â&#x201E;&#x2DC;(z) 24 â&#x201E;&#x2DC;(z) â&#x201E;&#x2DC;(z) 24 â&#x201E;&#x2DC;(z) where â&#x201E;&#x2DC; satisfies (â&#x201E;&#x2DC; )2 = 4â&#x201E;&#x2DC;3 â&#x2C6;&#x2019; 8. â&#x2C6;&#x161; If we continue from step 3 applying the change of variables (2x,â&#x2C6;&#x161; 23 y) we obtain the curve E3 : y 2 = 4x3 â&#x2C6;&#x2019; 1 and the map ÎŚ = ÎŚâ&#x2C6;&#x2019;1 (2x, 23 y) : E3 â&#x2020;&#x2019; F3 â&#x2C6;&#x161; (14) ÎŚ(x, y) = ÎŚâ&#x2C6;&#x2019;1 (2x, 23 y)

â&#x2C6;&#x161; â&#x2C6;&#x161; 23 y 1 23 y 1 â&#x2C6;&#x2019; â&#x2C6;&#x161; + â&#x2C6;&#x161; = , 2x 2 24x 2x 2 24x

1 y y 1 = 1â&#x2C6;&#x2019; â&#x2C6;&#x161; 1+ â&#x2C6;&#x161; , , 2x 2x 3x 3x and taking Î&#x203A; associated to g2 = 0 and g3 = 1, and Ψ : C/Î&#x203A; â&#x2020;&#x2019; E3 as (9), composing this two isomorphism we obtain the biholomorphism expected in (3) ÎŚ â&#x2014;Ś Ψ : C/Î&#x203A; â&#x2020;&#x2019; F3 :

1 1

â&#x2C6;&#x2019;1/2 â&#x2C6;&#x2019;1/2 1â&#x2C6;&#x2019;3 1+3 ÎŚ â&#x2014;Ś Ψ (z) = â&#x201E;&#x2DC; (z) , â&#x201E;&#x2DC; (z) , 2â&#x201E;&#x2DC;(z) 2â&#x201E;&#x2DC;(z) where the Weierstrass elliptic function â&#x201E;&#x2DC; satisfies here (â&#x201E;&#x2DC; )2 = 4â&#x201E;&#x2DC;3 â&#x2C6;&#x2019; 1.

On the other hand, let Ď&#x20AC; : C â&#x2020;&#x2019; C/Î&#x203A; be the natural projection, this map is an unbranched holomorphic covering, then the map ÎŚ â&#x2014;Ś Ψ â&#x2014;Ś Ď&#x20AC; : C â&#x2020;&#x2019; F3 is an unbranched holomorphic covering as well. Hence,

Baker-Gross theorem

23

given F and G a meromorphic solution of the Fermat cubic, the map φ(z) = (F (z), G(z)) defines a holomorphic map φ : C → F3 . Since C is simply connected φ has an holomorphic lifting α : C → C with respect to this covering, i.e., the following diagram commutes: C

(15) α

C

φ

Φ◦Ψ ◦π

F3

Composing with α we obtain

1 1 − 3−1/2 ℘ (α) , (16) F = 2℘(α)

G=

1 1 + 3−1/2 ℘ (α) , 2℘(α)

which are the desired formulas. This proves the theorem.

Note that we could use the map Φ−1 ◦ Ψ : C/Λ → F3 given in (13) instead of Φ ◦ Ψ in the above argument to obtain that any meromorphic solution of the Fermat cubic is of the form

1 1 1 1 (17) F = 1 − √ ℘ (α) , G = 1 + √ ℘ (α) , ℘(α) ℘(α) 24 24 where in this case ℘ satisfies (℘ )2 = 4℘3 − 8. We could obtain similar solutions depending on which factor we choose in step 3, but we can always obtain one from the other by this process.

3

Some remarks for Fermat curves of higher degree.

We finalize discussing about the application of the changes of variables described in 1.2 to the Fermat curves of higher degrees (see (18)). When the curve is of odd degree the process give us directly an interesting equation, but when the degree is even we need to apply a slight modification in step 1. From these equations we give a meromorphic function on the Fermat curves.

3.1

The odd case

The changes of variables in steps 1 and 2 described in 1.2 can be applied to any Fermat curve, (18)

Fn : xn + y n = 1,

24

José Juan-Zacarı́as

but in the case of n odd we get an interesting formula. By a straightforward calculation, following steps 1 and 2, we find the curve E2 :

2

n n−1

(19)

E2 : 2 + 2

y 2k = xn .

2k

k=1

As we did not modify the above steps we get the same correspondence Φ : Fn → E2 as in (12) but without step 3, so we get in this case:

y−x 2 (20) , , Φ(x, y) = y+x y+x

1 y 1 y Φ−1 (x, y) = − , + . x x x x Note that E2 has an holomorphic involution I(x, y) = (x, −y). It is easy to check that it is conjugate by Φ to the canonical involution of Fn , I(x, y) = (y, x), i.e., the following diagram commutes (21)

I

F3

F3

Φ

Φ

I

E2

E2

Note that the projection in the first coordinate is a meromorphic function of degree n − 1 on E2 , so composing with Φ we obtain the meromorphic function 2/(y + x) on Fn of degree n − 1, for example in the case n = 3 we obtain a degree 2 meromorphic function on the elliptic curve F3 .

3.2

The even case

Similar formulas can be obtained in the even case by using the change (x + ωy, x + y) instead of (x − y, x + y) in the first step, where ω is a root of xn = −1, maintaining the other steps without changes as before. In this case we have (22)

E2 : 2 +

n−1

k=1

n (1 + ω k )y k = xn , k

Baker-Gross theorem

and Φ : Fn → E2 become (23)

25

ω−1 x−y , , Φ(x, y) = ωy − x ωy − x

y 1 y 1 Φ−1 (x, y) = +ω , + . x x x x

Similarly as above, the map (ω − 1)/(ωy − x) is an meromorphic map of degree n − 1 on the Fermat curve Fn , for n even.

Acknowledgment I would like to thank my advisor Alberto Verjovsky for his constant support and for his encouragement in writing this paper. Also, I would like to thank the referee for his valuable comments which helped to improve the paper.

Instituto de Matemáticas Unidad Cuernavaca, Universidad Nacional Autónoma de México, Av. Universidad s/n. Col. Lomas de Chamilpa Código Postal 62210, Cuernavaca, Morelos. jose.juan@im.unam.mx

References [1] Forster O., Lectures on Riemann surfaces, Graduate Texts in Mathematics, 81, Springer-Verlag (1981).

[2] Gross F., On the equation f n + g n = 1, Bull. Amer. Math. Soc., 72 (1966), 86–88.

[3] Gross F., Erratum: On the equation f n + g n = 1, Bull. Amer. Math. Soc., 72 (1966), 576.

[4] Baker I. N., On a class of meromorphic functions, Proc. Amer. Math. Soc., 17 (1966), 819–822.

[5] Fred Gross, On the equation f n + g n = 1. II, Bull. Amer. Math. Soc., 74 (1968), 647–648.

[6] Joseph H. Silverman, The arithmetic of elliptic curves, Second edition, Graduate texts in mathematics, 106, Springer-Verlag (2009).

[7] Norbert Steinmetz, Nevanlinna theory, Normal families, and algebraic differential equations, Universitext, Springer (2017).

Morfismos, Vol. 22, No. 1, 2018, pp. 27–39 Morfismos, Vol. 22, No. 1, 2018, pp. 27–39

Non-contractible configuration spaces Non-contractible configuration spaces Cesar A. Ipanaque Zapata 1 Cesar A. Ipanaque Zapata

1

Abstract Let F (M, k) be the configuration space of ordered k−tuples of distinct points in the manifoldAbstract M . Using the Fadell-Neuwirth fibration, we prove that the configuration spaces (M, k) are never of Let F (M, k) be the configuration space ofFordered k−tuples contractible, for k ≥ 2.

As applications of our results, we will distinct points in the manifold M . Using the Fadell-Neuwirth calculate the LS category and topological complexity for its loop fibration, we prove that the configuration spaces F (M, k) are never space and suspension. contractible, for k ≥ 2. As applications of our results, we will calculate the LS category and topological complexity for its loop

2010 Mathematics Subject Classification: 55R80, 55S40, 55P35 (prispace and suspension. mary ) 55M30 (secondary).

Keywords and phrases: Subject OrderedClassification: configuration 55R80, spaces, 55S40, Fadell-Neuwirth 2010 Mathematics 55P35 (prifibration, spaces, suspension, Lusternik-Schnirelmann catmary )pointed 55M30loop (secondary). egory, Topological Keywords and complexity. phrases: Ordered configuration spaces, Fadell-Neuwirth fibration, pointed loop spaces, suspension, Lusternik-Schnirelmann category, Topological complexity.

1

Introduction

Let 1 X beIntroduction the space of all possible configurations or states of a mechanical system. A motion planning algorithm on X is a function which assigns to any configurations (A, B) configurations ∈ X × X, an initial state and a Let pair X beofthe space of all possible or states of aAmechanical desired stateAB,motion a continuous motion of theon system at which the initial system. planning algorithm X is starting a function assigns stateto Aany and ending at the desired state B. The elementary problem pair of configurations (A, B) ∈ X × X, an initial state A and a of robotics, the motion planning motion problem, of starting finding aatmotion desired state B, a continuous of consists the system the initial planning for a given system.

The elementary motion planning state algorithm A and ending at themechanical desired state B. The problem algorithm shouldthe bemotion continuous, that problem, is, it depends continuously the of robotics, planning consists of findingon a motion pairplanning of pointsalgorithm (A, B). Absence of continuity result The in the instability for a given mechanicalwill system. motion planning of behavior of should the motion planning. Unfortunately, a continuous motion algorithm be continuous, that is, it depends continuously on the 1 pair of points (A, B). Absence of continuity will result in the instability This work is a part of my PhD’s thesis under the supervision of professor of behavior of the theUniversidade motion planning.

Unfortunately, a continuous motion Denise de Mattos at de São Paulo and it is supported

by FAPESP 2016/18714-8.

1 This work is a part of my PhD’s thesis under the supervision of professor Denise de Mattos at the Universidade de São Paulo and it is supported by FAPESP 2016/18714-8.

27

27

28

Cesar A. Ipanaque Zapata

planning algorithm on space X exists if and only if X is contractible, see [10]. The design of effective motion planning algorithms is one of the challenges of modern robotics, see, for example Latombe [18] and LaValle [19].

Investigation of the problem of simultaneous motion planning without collisions for k robots in a topological manifold M leads one to study the (ordered) configuration space F (M, k). We want to know if exists a continuous motion planning algorithm on the space F (M, k). Thus, an interesting question is whether F (M, k) is contractible.

It seems likely that the configuration space F (M, k) is not contractible for certain topological manifolds M . Evidence for this statement is given in the work of F. Cohen and S. Gitler, in [4], they described the homology of loop spaces of the configuration space F (M, k) whose results showed that this homology is non trivial. In a robotics setting, the (collision-free) motion planning problem is challenging since it is not known an effective motion planning algorithm, see [20].

In this paper, using the Fadell-Neuwirth fibration, we will prove that the configuration spaces F (M, k) of topological manifolds M , are never contractible (see Theorem 2.1). Note that the configuration space F (X, k) can be contractible, for any k ≥ 1 (e.g. if X is an infinite indiscrete space or if X = R∞ ). As applications of our results, we will calculate the LS category and topological complexity for the (pointed) loop space ΩF (M, k) (see Theorem 4.7) and the suspension ΣF (M, k) (see Theorem 4.11 and Proposition 4.17).

Conjecture 1.1. If X is a path-connected and paracompact topological space with covering dimension 1 ≤ dim(X) < ∞. Then the configuration spaces F (X, k) are never contractible, for k ≥ 2.

Computation of LS category and topological complexity of the configuration space F (M, k) is a great challenge. The LS category of the configuration space F (Rm , k) has been computed by Roth in [21]. In Farber and Grant’s work [11], the authors computed the TC of the configuration space F (Rm , k). Farber, Grant and Yuzvinsky determined the topological complexity of F (Rm − Qr , k) for m = 2, 3 in [12]. Later González and Grant extended the results to all dimensions m in [15].

Cohen and Farber in [2] computed the topological complexity of the configuration space F (Σg − Qr , k) of orientable surfaces Σg . Recently in [24], the author computed the LS category and TC of the configuration space F (CPm , 2). The LS category and TC of the configuration space of ordered 2−tuples of distinct points in G × Rn has been computed by

Non-contractible configuration spaces

29

the author in [25]. Many more related results can be found in the recent survey papers [1] and [9].

2

Main Results

Let M denote a connected m−dimensional topological manifold (without boundary), m ≥ 1. The configuration space F (M, k) of ordered k−tuples of distinct points in M (see [8]) is the subspace of M k given by F (M, k) = {(m1 , . . . , mk ) ∈ M k | mi = mj , ∀i = j}.

Let Qr = {q1 , . . . , qr } denote a set of r distinct points of M .

Let M be a connected finite dimensional topological manifold (without boundary) with dimension at least 2 and k > r ≥ 1. It is well known that the projection map (1)

πk,r : F (M, k) −→ F (M, r), (x1 , . . . , xk ) → (x1 , . . . , xr )

is a fibration with fibre F (M −Qr , k−r). It is called the Fadell-Neuwirth fibration [6]. In contrast, when the manifold M has nonempty boundary, πk,r is not a fibration. The fact that the map πk,r is not a fibration may be seen by considering, for example, the manifold M = D2 that is with boundary but the fibre D2 − {(0, 0)} is not homotopy equivalent to the fibre D2 − {(1, 0)}.

Let X be a space, with base-point x0 . The pointed loop space is denoted by ΩX, as its base-point, if it needs one, we take the function w0 constant at x0 . We recall that a topological space X is weak-contractible if all homotopy groups of X are trivial, that is, πn (X, x0 ) = 0 for all n ≥ 0 and all choices of base point x0 .

In this paper, using the Fadell-Neuwirth fibration, we prove the following theorem Theorem 2.1. [Main Theorem] If M is a connected finite dimensional topological manifold, then the configuration space F (M, k) is not contractible (indeed, it is never weak-contractible), for any k ≥ 2.

Remark 2.2. Theorem 2.1 can be proved using classifying spaces. I am very grateful to Prof. Nick Kuhn for his suggestion about the following proof. Let M be a connected finite dimensional topological manifold.

If the configuration space F (M, k) was contractible, then the quotient F (M, k)/Sk would be a finite dimensional model for the classifying space

30

Cesar A. Ipanaque Zapata

of the k th symmetric group Sk . But if G is a nontrivial finite group or even just contains any nontrivial elements of finite order, then there is no finite dimensional model for BG because H â&#x2C6;&#x2014; (G) is periodic. Thus F (M, k) is never contractible for k â&#x2030;Ľ 2.

3

PROOF of Theorem 2.1

The proof of Theorem 2.1 is greatly simplified by actually working on two main steps: S1. We first get the Theorem 2.1 when Ď&#x20AC;1 (M ) = 0 (Proposition 3.5).

S2. Then we prove the Theorem 2.1 when Ď&#x20AC;1 (M ) = 0 (It follows from Lemma 3.6).

Here we note that the manifolds being considered are without boundary.

Step S1 above is accomplished proving the next four results.

Lemma 3.1. Let M denote a connected mâ&#x2C6;&#x2019;dimensional topological manifold, m â&#x2030;Ľ 2. If r â&#x2030;Ľ 1, then the configuration space F (M â&#x2C6;&#x2019; Qr , k) is not contractible (indeed, it is not weak-contractible), â&#x2C6;&#x20AC;k â&#x2030;Ľ 2.

Proof. Recall that if p : E â&#x2C6;&#x2019;â&#x2020;&#x2019; B is the projection map in a fibration with inclusion of the fibre i : F â&#x2C6;&#x2019;â&#x2020;&#x2019; E such that p supports a crosssection Ď&#x192;, then (1) Ď&#x20AC;q (E) â&#x2C6;ź = Ď&#x20AC;q (F ) â&#x160;&#x2022; Ď&#x20AC;q (B), â&#x2C6;&#x20AC;q â&#x2030;Ľ 2 and (2) Ď&#x20AC;1 (E) â&#x2C6;ź = Ď&#x20AC;1 (F ) Ď&#x20AC;1 (B).

If r â&#x2030;Ľ 1, then the first coordinate projection map Ď&#x20AC; : F (M â&#x2C6;&#x2019; Qr , k) â&#x2C6;&#x2019;â&#x2020;&#x2019; M â&#x2C6;&#x2019; Qr is a fibration with fibre F (M â&#x2C6;&#x2019; Qr+1 , k â&#x2C6;&#x2019; 1) and Ď&#x20AC; admits a section ([8], kâ&#x2C6;&#x2019;1 Theorem 1). Thus (1) Ď&#x20AC;q (F (M â&#x2C6;&#x2019; Qr , k)) â&#x2C6;ź = i=0 Ď&#x20AC;q (M â&#x2C6;&#x2019; Qr+i ), â&#x2C6;&#x20AC;q â&#x2030;Ľ 2 ([8], Theorem 2) and (2) Ď&#x20AC;1 (F (M â&#x2C6;&#x2019; Qr , k)) is isomorphic to

¡ ¡ ¡ Ď&#x20AC;1 (M â&#x2C6;&#x2019; Qr+kâ&#x2C6;&#x2019;1 ) Ď&#x20AC;1 (M â&#x2C6;&#x2019; Qr+kâ&#x2C6;&#x2019;2 ) ¡ ¡ ¡

( Ď&#x20AC;1 (M â&#x2C6;&#x2019; Qr+1 ) Ď&#x20AC;1 (M â&#x2C6;&#x2019; Qr ) Finally, notice that M â&#x2C6;&#x2019; Qr+kâ&#x2C6;&#x2019;1 is homotopy equivalent to r+kâ&#x2C6;&#x2019;2

i=1

Smâ&#x2C6;&#x2019;1 â&#x2C6;¨ (M â&#x2C6;&#x2019; V ),

Non-contractible configuration spaces

31

where V is an open m−ball in M such that Qr+k−1 ⊂ V ([7], Proposition 3.1). Thus M −Qr+k−1 is not weak contractible, therefore F (M −Qr , k) is not weak-contractible.

Lemma 3.2. If M is a simply-connected finite dimensional topological manifold which is not weak-contractible, then the singular homology (with coefficients in a field K) of ΩM does not vanish in sufficiently large degrees.

Proof. By contradiction, we will suppose the singular homology of ΩM vanishes in sufficiently large degrees, that is, there exists an integer q0 ≥ 1 such that, Hq (ΩM ; K) = 0, ∀q ≥ q0 , where K is a field. Let f denote a nonzero homology class of maximal degree in H∗ (ΩM ; K). As M is finite dimensional and not weak-contractible, let b denote a nonzero ∗ (−; K) denote ∗ (M ; K) of maximal degree (here H homology class in H reduced singular homology, with coefficients in a field K). Notice that b ⊗ f survives to give a non-trivial class in the Serre spectral sequence abutting to H∗ (P (M, x0 ); K), since M is simply-connected, the local coefficient system H∗ (ΩM ; K) is trivial, where P (M, x0 ) = {γ ∈ P M | γ(0) = x0 }, it is contractible. This is a contradiction and so the singular homology of ΩM does not vanish in sufficiently large degrees.

Proposition 3.3. Let M be a simply-connected topological manifold which is not weak-contractible with dimension at least 2. Then the configuration space F (M, k) is not contractible (indeed, it is never weakcontractible), ∀k ≥ 2.

Proof. By hypothesis, M is a connected finite dimensional topological manifold of dimension at least 2. Consequently, there is a fibration F (M, k) −→ M with fibre F (M − Q1 , k − 1) (k ≥ 2). We just have to note that in sufficiently large degrees, the singular homology, with coefficients in a field K, of F (M − Q1 , k − 1) vanishes, since F (M − Q1 , k − 1) is a connected finite dimensional topological manifold.

On the other hand, if F (M, k) were weak-contractible, then the pointed loop space of M is weakly homotopy equivalent to F (M − Q1 , k − 1) which it cannot be by Lemma 3.2. Thus, the configuration space F (M, k) is not weak-contractible.

32

Cesar A. Ipanaque Zapata

Proposition 3.4. Let M be a weak-contractible topological manifold with dimension at least 2. Then the configuration space F (M, k) is not contractible (indeed, it is never weak-contractible), ∀k ≥ 2.

Proof. By the homotopy long exact sequence of the fibration F (M, k) −→ M with fibre F (M − Q1 , k − 1), we can conclude the inclusion i : F (M − Q1 , k − 1) → F (M, k) is a weak homotopy equivalence. If k ≥ 3, then Lemma 3.1 implies that F (M − Q1 , k − 1) is not weak-contractible and so F (M, k) is not weak-contractible. If k = 2, we consider the cover M = A ∪ B, where A = M − {q}, B = M − {q }, q, q distinct. Here we note that A = M − {q} and B = M − {q } are homeomorphic to M − Q1 and A ∩ B = M − {q, q } is not weak-contractible, because M − {q, q } is homotopy equivalent to the wedge Sm−1 ∨ (M − V ), where V is an open m−ball in M such that {q, q } ⊂ V ([7], Proposition 3.1). Thus, the Mayer-Vietoris sequence, for the given cover, implies M − Q1 is not weak-contractible and so F (M, 2) is not weak-contractible. Therefore, F (M, k) is not weak-contractible.

By Propositions 3.3 and 3.4 we have the following statement.

Proposition 3.5. If M is a simply-connected topological manifold with dimension at least 2, then the configuration space F (M, k) is not contractible (indeed, it is never weak-contractible), ∀k ≥ 2.

A key ingredient for step S2 is given by the next result.

Lemma 3.6. If M is a connected finite dimensional topological manifold with dimension at least 2, then the inclusion map i : F (M, k) −→ M k induces a homomorphism i∗ : π1 F (M, k) −→ π1 M k which is surjective.

Proof. We will prove it by induction on k. We just have to note that the inclusion map j : M − Qk −→ M induces an epimorphism j∗ : π1 (M − Qk ) −→ π1 M, for any k ≥ 1. The following diagram of fibrations (see Figure 1) is commutative.

20/03/2017

Preview

Non-contractible configuration spaces

33

Figure 1: Commutative diagram.

Thus by induction, we can conclude the inclusion map i : F (M, k) −→ M k induces a homomorphism i∗ : π1 F (M, k) −→ π1 M k which is surjective and so we are done.

Remark 3.7. Lemma 3.6 is actually a very special case of a general theorem of Golasiński, Gonçalves and Guaschi in ([13], Theorem 3.2).

Also, it can be proved using braids ([14], Lemma 1).

Proof of Theorem 2.1. The case dim M = 1 is straightforward, so we assume that dim M ≥ 2. If π1 (M ) = 0 then the result follows easily from the Proposition 3.5. If π1 (M ) = 0 then π1 (M k ) = 0 and by Lemma 3.6 i∗ : π1 (F (M, k)) −→ π1 (M k ) is an epimorphism. Thus π1 (F (M, k)) = 0 and F (M, k) is not weakcontractible. Therefore, F (M, k) is not contractible.

4

Lusternik-Schnirelmann category and topological complexity

As applications of our results, in this section, we will calculate the LS category and topological complexity for the (pointed) loop space ΩF (M, k) and the suspension ΣF (M, k).

Here we follow a definition of category, one greater than category given in [5].

Definition 4.1. We say that the Lusternik-Schnirelmann category or category of a topological space X, denoted cat(X), is the least integer m such that X can be covered with m open sets, which are all contractible within X. If no such m exists we will set cat(X) = ∞.

34

Cesar A. Ipanaque Zapata

Let P X denote the space of all continuous paths γ : [0, 1] −→ X in X and π : P X −→ X × X denotes the map associating to any path γ ∈ P X the pair of its initial and end points π(γ) = (γ(0), γ(1)). Equip the path space P X with the compact-open topology.

Definition 4.2. [10] The topological complexity of a path-connected space X, denoted by T C(X), is the least integer m such that the Cartesian product X × X can be covered with m open subsets Ui , X × X = U1 ∪ U2 ∪ · · · ∪ Um such that for any i = 1, 2, . . . , m there exists a continuous function si : Ui −→ P X, π ◦ si = id over Ui . If no such m exists we will set T C(X) = ∞.

Remark 4.3. For all path connected spaces X, the basic inequality that relate cat and T C is cat(X) ≤ T C(X).

On the other hand, by ([10], Theorem 5), for all path connected paracompact spaces X, T C(X) ≤ 2cat(X) − 1.

It follows from the Definition 4.1 that we have cat(X) = 1 if and only if X is contractible. It is also easy to show that T C(X) = 1 if and only if X is contractible.

By Remark 4.3 and Theorem 2.1, we obtain the following statement.

Proposition 4.4. If M is a connected finite dimensional topological manifold, then the Lusternik-Schnirelmann category and the topological complexity of F (M, k) are at least 2, ∀k ≥ 2.

Proposition 4.5 and Lemma 4.6 we state in this section are known, they can be found in the paper by Frederick R. Cohen [3]. Here Ωj0 X denotes the component of the constant map in the j th pointed loop space of X. Proposition 4.5. ([3], Theorem 1) If X is a simply-connected finite complex which is not contractible, then the Lusternik-Schnirelmann category of Ωj0 X is infinite for j ≥ 1.

Non-contractible configuration spaces

35

Lemma 4.6. Let M be a simply-connected finite dimensional topological manifold with dimension at least 3. If M has the homotopy type of a finite CW complex, then the configuration space F (M, k) has the homotopy type of a finite CW complex, ∀k ≥ 1.

As a consequence of Theorem 2.1 we can obtain Proposition 4.5 for configuration spaces.

Theorem 4.7. Let M be a space which has the homotopy type of a finite CW complex. If M is a simply-connected finite dimensional topological manifold with dimension at least 3, then the Lusternik-Schnirelmann category and the topological complexity of Ωj0 F (M, k) are infinite, for any k ≥ 2 and j ≥ 1.

Proof. The assumptions that M is a simply-connected finite dimensional topological manifold with dimension at least 3, imply the configuration space F (M, k) is simply-connected. Furthermore, as M has the homotopy type of a finite CW complex, the configuration space F (M, k) also has the homotopy type of a finite CW complex by Lemma 4.6.

Finally the configuration space F (M, k) is not contractible by Theorem 2.1. Therefore we can apply Proposition 4.5 and conclude that the Lusternik-Schnirelmann category of Ωj0 F (M, k) is infinite, ∀k ≥ 2.

Moreover, by Remark 4.3, the topological complexity of Ωj0 F (M, k) is also infinite, ∀k ≥ 2.

Remark 4.8.

1. In Theorem 4.7, the assumption M has the homotopy type of a finite CW complex can be reduce to the assumption M is a CW complex of finite type (see [22]).

2. By Theorem 4.7, if G is a simply-connected finite dimensional Lie group of finite type with dimension at least 3. Then the topological complexity T C(ΩF (G, k)) = ∞, for any k ≥ 2. In contrast, we will see that the topological complexity T C(ΣF (G, k)) = 3 < ∞, for any k ≥ 3.

Remark 4.9. If X is any topological space and ΣX :=

X × [0, 1] X × {0} ∪ X × {1}

is the non-reduced suspension of the space X, it is well-known that cat(ΣX) ≤ 2. We can cover ΣX by two overlapping open sets (e.g, q(X × [0, 3/4) and q(X × (1/4, 1]), where q : X × [0, 1] −→ ΣX is

36

Cesar A. Ipanaque Zapata

the projection map), such that each open set is homeomorphic to the cone CX := X×[0,1] X×{0} , so they are contractible in itself and thus they are contractible in the suspension ΣX. Lemma 4.10. Let X be a simply-connected topological space. If X is not weak-contractible, then cat(ΣX) = 2.

Proof. It is sufficient to prove that ΣX is not weak-contractible and thus cat(ΣX) ≥ 2. Since contractible implies weak-contractible. If ΣX was weak-contractible then by the Mayer-Vietoris sequence for the open covering ΣX = q(X × [0, 3/4) ∪ q(X × (1/4, 1]) we can conclude Hq (X; Z) = 0, ∀q ≥ 1. Thus by ([17], Corollary 4.33) X is weak-contractible (here we have used that X is simply-connected2 ). It is a contradiction with the hypothesis. Therefore ΣX is not weakcontractible.

Theorem 4.11. If M is a simply-connected finite dimensional topological manifold with dimension at least 3, then cat(ΣF (M, k)) = 2, ∀k ≥ 2.

Proof. The arguments M is a simply-connected finite dimensional topological manifold with dimension at least 3, imply the configuration space F (M, k) is simply-connected. The configuration space F (M, k) is not weak-contractible by Theorem 2.1. Therefore we can apply Lemma 4.10 and the Lusternik-Schnirelmann category of ΣF (M, k)) is two, ∀k ≥ 2.

We note that ΣF (M, k) is paracompact because F (M, k) is paracompact.

Corollary 4.12. If M is a simply-connected finite dimensional topological manifold with dimension at least 3, then 2 ≤ T C(ΣF (M, k)) ≤ 3, ∀k ≥ 2.

Proof. It follows from Remark 4.3 and Theorem 4.11.

Remark 4.13. By Corollary 4.12 the topological complexity of the suspension of a configuration space is secluded in the range 2 ≤ T C(ΣF (M, k)) ≤ 3 2

By Hatcher ([17], Example 2.38) there exists nonsimply-connected acyclic spaces.

Non-contractible configuration spaces

37

and any value in between can be taken (e.g. if M = Sm or Rm and k = 2).

Now we will recall the definition of the cup-length.

Definition 4.14. [5] Let R be a commutative ring with unit and X be a topological space. The cup-length of X, denote cupR (X), is the least integer n such that all (n + 1)−fold cup products vanish in the reduced (X; R). cohomology H Remark 4.15. ([5], Theorem 1.5) Let R be a commutative ring with unit and X be a topological space. It is well-known that 1 + cupR (X) ≤ cat(X).

On the other hand, it is easy to verify that the cup-length has the property listed below.

Lemma 4.16. Let K be a field and X, Y be topological spaces. Then if H k (Y ; K) is a finite dimensional K−vector space for all k ≥ 0. We have cupK (X × Y ) ≥ cupK (X) + cupK (Y ).

Proposition 4.17. If G is a simply-connected finite dimensional Lie group of finite type with dimension at least 3. Then T C(ΣF (G, k)) = 3, ∀k ≥ 3.

Proof. We will assume that G is not contractible, the case G is contractible follows easily because F (G, k) is homotopy equivalent to the configuration space F (Rd , k), where d = dim(G) (see [23], pg. 118).

By Corollary 4.12 it is sufficient to prove that T C(ΣF (G, k)) = 2. If T C(ΣF (G, k)) = 2 then, by ([16], Theorem 1), we have ΣF (G, k) is homotopy equivalent to some (odd-dimensional) sphere. Then F (G, k) is homotopy equivalent to some (even-dimensional) sphere and thus cat(F (G, k)) = 2. On the other hand, F (G, k) is homeomorphic to the product G × F (G − {e}, k − 1) because G is a topological group.

Then 2 = cat(G × F (G − {e}, k − 1)) ≥ cupK (G × F (G − {e}, k − 1)) + 1 for any field K (see Remark 4.15). Furthermore, Lemma 4.16 implies that cupK (G × F (G − {e}, k − 1)) ≥ cupK (G) + cupK (F (G − {e}, k − 1)) ≥ 1+1 = 2

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Cesar A. Ipanaque Zapata

(here we note that k −1 ≥ 2 and by Theorem 2.1 we have the cup-length cupK (F (G − {e}, k − 1)) ≥ 1). Thus, 2 = cat(G × F (G − {e}, k − 1)) ≥ 3 which is a contradiction.

Acknowledgement The author is very grateful to Frederick Cohen and Jesús González for their comments and encouraging remarks which were of invaluable mental support.

Cesar A. Ipanaque Zapata Departamento de Matemática, Universidade de São Paulo, Instituto de ciências matemáticas e de computação - USP, Avenida Trabalhador São-carlense, 400 - Centro CEP: 13566-590 - São Carlos - SP, Brasil, cesarzapata@usp.br

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Morfismos se imprime en el taller de reproducción del Departamento de Matemáticas del Cinvestav, localizado en Avenida Instituto Politécnico Nacional 2508, Colonia San Pedro Zacatenco, C.P. 07360, México, D.F. Este número se terminó de imprimir en el mes de agosto de 2018. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

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