Morfismos, Vol 21, No 1, 2017

VOLUMEN 21 NÚMERO 1 ENERO A JUNIO DE 2017 ISSN: 1870-6525

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VOLUMEN 21 NÚMERO 1 ENERO A JUNIO DE 2017 ISSN: 1870-6525

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Morfismos

Contents - Contenido An approach to the topological complexity of the Klein bottle Donald M. Davis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The stability theorem of persistent homology Adam Gardner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Aproximacio ´n m´etrica de grupos: una breve perspectiva Luis Manuel Rivera y Nidya Monserrath Veyna Garc´ıa . . . . . . . . . . . . . . . . . . 45

Morfismos, Vol. 21, No. 1, 2017, pp. 1–13 Morfismos, Vol. 21, No. 1, 2017, pp. 1–13

An approach to the topological complexity of the bottle complexity of the An approach to Klein the topological Klein bottle Donald M. Davis Donald M. Davis Abstract Recently, Cohen and Vandembroucq proved that the reduced topological complexity of the KleinAbstract bottle is 4. Simultaneously and independently we announced a proof of the same result.

Mistakes Recently, Cohen and Vandembroucq proved that the reduced topowerelogical found complexity in our argument, quiteisdifferent than theirs. and of thewhich Kleinwas bottle 4. Simultaneously After correcting these, we found athat our of theMistakes obindependently we announced proof of description the same result. struction class agreed with theirs.

Our was approach to showing that were found in our argument, which quite different than theirs. this After obstruction is nonzero to dothat so, while theirs did not correcting these,failed we found our description of fail. the obHerestruction we discuss ouragreed approach, directly with thethat class withwhich theirs.deals Our more approach to showing simplicial structure of the Klein bottle. this obstruction is nonzero failed to do so, while theirs did not fail.

Here we discuss our approach, which deals more directly with the 2010 Mathematics Subject Classification: 55M30, 55N25, 57M20. simplicial structure of the Klein bottle.

Keywords and phrases: Topological complexity, Klein bottle, obstruction 2010 Mathematics Subject Classification: 55M30, 55N25, 57M20. theory.

Keywords and phrases: Topological complexity, Klein bottle, obstruction theory.

1

Introduction

The1reduced topological complexity, TC(X), of a topological space X, Introduction as introduced in [5], is roughly one less than the minimal number of reducedtotopological TC(X), a topological space rulesThe required tell how tocomplexity, move between anyoftwo points of X. A X, as introduced in [5], roughly less than thecontinuously minimal number “rule” must be such thatisthe choiceone of path varies with of required tell how to move has between two points of X. A the rules endpoints.

An to outstanding problem been any to determine TC(K), “rule” be such that the choice of path varies continuously with where K ismust the Klein bottle. theaendpoints.

An outstanding problem has been to determine In recent preprint ([1]), Cohen and Vandembroucq proved TC(K), that where K is the Klein bottle.

TC(K) = 4. Simultaneously and independently we announced a proof In a recent Vandembroucq proved of the same result. preprint Mistakes([1]), wereCohen found and in our argument, which wasthat TC(K) = 4.than Simultaneously independently proof quite different theirs. Afterand correcting these, we announced found that aour of the same result.

Mistakes were found in our argument, which was description of the obstruction class agreed with theirs.

Our approach quite different than theirs. After correcting we while found theirs that our to showing that this obstruction is nonzero failedthese, to do so, description of the obstruction class agreed with theirs. Our approach to showing that this obstruction1 is nonzero failed to do so, while theirs 1

2

Donald M. Davis

did not fail. Here we discuss our approach, which deals more directly with the simplicial structure of the Klein bottle.

The Klein bottle is homeomorphic to the space of all configurations of various physical systems. For example, it is homeomorphic to the space of all planar 5-gons with side lengths 1, 1, 2, 2, and 3 ([7, Table B]). Such a polygon can be considered as linked robot arms. Knowing that TC(K) = 4 implies that five rules are required to program these arms to move from any configuration to any other.

Our main tool is a result of Costa and Farber ([2]), which we state later as Theorem 4.1, which describes a single obstruction in H 2n (X Ă&#x2014; X; G), where G is a certain local coefficient system, for an n-dimensional cell complex X to satisfy TC(X) < 2n. We present an approach to proving that this class is nonzero for the Klein bottle K, which would imply that TC(K) â&#x2030;Ľ 4, while TC(K) â&#x2030;¤ 4 for dimensional reasons.([5, Cor 4.15])

2

The â&#x2C6;&#x2020;-complex for K Ă&#x2014; K

A â&#x2C6;&#x2020;-complex, as described in [6], is essentially a quotient of a simplicial complex, with certain simplices identified. As discussed there, this notion is equivalent to that of semi-simplicial complex introduced in [3].

It is important that vertices be numbered prior to identifications, and that simplices be described by writing vertices in increasing order.

The â&#x2C6;&#x2020;-complex that we will use for K is given below. It has one vertex v, three edges, (0, 2) = (4, 5), (1, 2) = (3, 4), and (0, 1) = (3, 5), and two 2-cells, (0, 1, 2) and (3, 4, 5). a

5 1

4

b

b

c

3 0

a

2

If K and L are simplicial complexes with an ordering of the vertices of each, then the simplices of the simplicial complex K Ă&#x2014; L are all

Approach to TC of Klein bottle

3

(vi0 , wj0 ), . . . , (vik , wjk ) such that i0 ≤ · · · ≤ ik and j0 ≤ · · · ≤ jk and {vi0 , . . . , vik } and {wj0 , . . . , wjk } are simplices of K and L, respectively.

Note that we may have vit = vit+1 or wjt = wjt+1 , but not both (for the same t). Now, if K and L are ∆-complexes, i.e., they have some simplices identified, then K × L has (vi0 , wj0 ), . . . , (vik , wjk ) ∼ (vi 0 , wj0 ), . . . , (vi k , wjk ) iff {vi0 , . . . , vik } ∼ {vi 0 , . . . , vi k } and {wj0 . . . wjk } ∼ {wj0 . . . wjk }, and the positions of the repetitions in (vi0 , . . . , vik ) and (vi 0 , . . . , vi k ) are the same, and so are those of (wj0 , . . . , wjk ) and (wj0 , . . . , wjk ). This description is equivalent to the one near the end of [4], called K∆L there. There is also a discussion in [6, pp.277-278].

Following this description, we now list the simplices of K ×K, where K is the above ∆-complex. We write v for the unique vertex when it is being producted with a simplex, but otherwise we list all vertices by their number. We omit commas in ordered pairs; e.g., 24 denotes the vertex of K × K which is vertex 2 in the first factor and vertex 4 in the second factor. There are 1, 15, 50, 60, and 24 distinct simplices of dimensions 0, 1, 2, 3, and 4, respectively. This is good since 1 − 15 + 50 − 60 + 24 = 0, the Euler characteristic of K × K. We number the simplices in each dimension, which will be useful later.

The only 0-simplex is (vv).

1-simplices: 1: (0v, 1v) = (3v, 5v).

2: (0v, 2v) = (4v, 5v).

3: (1v, 2v) = (3v, 4v).

4: (v0, v1) = (v3, v5).

5: (v0, v2) = (v4, v5).

6: (v1, v2) = (v3, v4).

7: (00, 11) = (30, 51) = (33, 55) = (03, 15).

8: (00, 12) = (30, 52) = (34, 55) = (04, 15).

9: (01, 12) = (31, 52) = (33, 54) = (03, 14).

10: (00, 21) = (40, 51) = (43, 55) = (03, 25).

11: (00, 22) = (40, 52) = (44, 55) = (04, 25).

12: (01, 22) = (41, 52) = (43, 54) = (03, 24).

13: (10, 21) = (30, 41) = (33, 45) = (13, 25).

14: (10, 22) = (30, 42) = (34, 45) = (14, 25).

15: (11, 22) = (31, 42) = (33, 44) = (13, 24).

4

Donald M. Davis

2-simplices 1: (0v, 1v, 2v).

2: (3v, 4v, 5v).

3: (v0, v1, v2).

4: (v3, v4, v5).

5: (00, 10, 11) = (30, 50, 51) = (03, 13, 15) = (33, 53, 55).

6: (00, 10, 12) = (30, 50, 52) = (04, 14, 15) = (34, 54, 55).

7: (01, 11, 12) = (31, 51, 52) = (03, 13, 14) = (33, 53, 54).

8: (00, 01, 11) = (30, 31, 51) = (03, 05, 15) = (33, 35, 55).

9: (00, 02, 12) = (30, 32, 52) = (04, 05, 15) = (34, 35, 55).

10: (01, 02, 12) = (31, 32, 52) = (03, 04, 14) = (33, 34, 54).

11: (00, 20, 21) = (40, 50, 51) = (03, 23, 25) = (43, 53, 55).

12: (00, 20, 22) = (40, 50, 52) = (04, 24, 25) = (44, 54, 55).

13: (01, 21, 22) = (41, 51, 52) = (03, 23, 24) = (43, 53, 54).

14: (00, 01, 21) = (40, 41, 51) = (03, 05, 25) = (43, 45, 55).

15: (00, 02, 22) = (40, 42, 52) = (04, 05, 25) = (44, 45, 55).

16: (01, 02, 22) = (41, 42, 52) = (03, 04, 24) = (43, 44, 54).

17: (10, 20, 21) = (30, 40, 41) = (13, 23, 25) = (33, 43, 45).

18: (10, 20, 22) = (30, 40, 42) = (14, 24, 25) = (34, 44, 45).

19: (11, 21, 22) = (31, 41, 42) = (13, 23, 24) = (33, 43, 44).

20: (10, 11, 21) = (30, 31, 41) = (13, 15, 25) = (33, 35, 45).

21: (10, 12, 22) = (30, 32, 42) = (14, 15, 25) = (34, 35, 45).

22: (11, 12, 22) = (31, 32, 42) = (13, 14, 24) = (33, 34, 44).

23: (00, 11, 22).

24: (30, 41, 52).

25: (03, 14, 25).

26: (33, 44, 55).

27: (00, 01, 22) = (40, 41, 52).

28: (30, 31, 52) = (00, 01, 12).

29: (03, 04, 25) = (43, 44, 55).

30: (33, 34, 55) = (03, 04, 15).

31: (00, 10, 22) = (04, 14, 25).

32: (30, 40, 52) = (34, 44, 55).

33: (03, 13, 25) = (00, 10, 21).

34: (33, 43, 55) = (30, 40, 51).

35: (00, 21, 22) = (40, 51, 52).

36: (03, 24, 25) = (43, 54, 55).

37: (30, 51, 52) = (00, 11, 12).

38: (33, 54, 55) = (03, 14, 15).

39: (00, 12, 22) = (04, 15, 25).

Approach to TC of Klein bottle

40: 41: 42: 43: 44: 45: 46: 47: 48: 49: 50:

(30, 42, 52) = (34, 45, 55).

(03, 15, 25) = (00, 11, 21).

(33, 45, 55) = (30, 41, 51).

(01, 12, 22) = (03, 14, 24).

(31, 42, 52) = (33, 44, 54).

(03, 13, 24) = (01, 11, 22).

(33, 43, 54) = (31, 41, 52).

(10, 21, 22) = (30, 41, 42).

(13, 24, 25) = (33, 44, 45).

(10, 11, 22) = (30, 31, 42).

(13, 14, 25) = (33, 34, 45).

3-simplices 1: (00, 10, 21, 22) 2: (00, 11, 21, 22).

3: (00, 10, 11, 22).

4: (00, 11, 12, 22).

5: (00, 01, 11, 22).

6: (00, 01, 12, 22).

7: (30, 40, 51, 52).

8: (30, 41, 51, 52).

9: (30, 40, 41, 52).

10: (30, 41, 42, 52).

11: (30, 31, 41, 52).

12: (30, 31, 42, 52).

13: (03, 13, 24, 25).

14: (03, 14, 24, 25).

15: (03, 13, 14, 25).

16: (03, 14, 15, 25).

17: (03, 04, 14, 25).

18: (03, 04, 15, 25).

19: (33, 43, 54, 55).

20: (33, 44, 54, 55).

21: (33, 43, 44, 55).

22: (33, 44, 45, 55).

23: (33, 34, 44, 55).

24: (33, 34, 45, 55).

25: (00, 10, 20, 21) = (03, 13, 23, 25).

26: (00, 10, 11, 21) = (03, 13, 15, 25).

27: (00, 01, 11, 21) = (03, 05, 15, 25).

5

6

Donald M. Davis

28: 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40: 41: 42: 43: 44: 45: 46: 47: 48: 49: 50: 51: 52: 53: 54: 55: 56: 57: 58: 59: 60:

(00, 10, 20, 22) = (04, 14, 24, 25).

(00, 10, 12, 22) = (04, 14, 15, 25).

(00, 02, 12, 22) = (04, 05, 15, 25).

(01, 11, 21, 22) = (03, 13, 23, 24).

(01, 11, 12, 22) = (03, 13, 14, 24).

(01, 02, 12, 22) = (03, 04, 14, 24).

(00, 01, 02, 12) = (30, 31, 32, 52).

(00, 01, 11, 12) = (30, 31, 51, 52).

(00, 10, 11, 12) = (30, 50, 51, 52).

(00, 01, 02, 22) = (40, 41, 42, 52).

(00, 01, 21, 22) = (40, 41, 51, 52).

(00, 20, 21, 22) = (40, 50, 51, 52).

(10, 11, 12, 22) = (30, 31, 32, 42).

(10, 11, 21, 22) = (30, 31, 41, 42).

(10, 20, 21, 22) = (30, 40, 41, 42).

(30, 40, 50, 51) = (33, 43, 53, 55).

(30, 40, 41, 51) = (33, 43, 45, 55).

(30, 31, 41, 51) = (33, 35, 45, 55).

(30, 40, 50, 52) = (34, 44, 54, 55).

(30, 40, 42, 52) = (34, 44, 45, 55).

(30, 32, 42, 52) = (34, 35, 45, 55).

(31, 41, 51, 52) = (33, 43, 53, 54).

(31, 41, 42, 52) = (33, 43, 44, 54).

(31, 32, 42, 52) = (33, 34, 44, 54).

(03, 04, 05, 15) = (33, 34, 35, 55).

(03, 04, 14, 15) = (33, 34, 54, 55).

(03, 13, 14, 15) = (33, 53, 54, 55).

(03, 04, 05, 25) = (43, 44, 45, 55).

(03, 04, 24, 25) = (43, 44, 54, 55).

(03, 23, 24, 25) = (43, 53, 54, 55).

(13, 14, 15, 25) = (33, 34, 35, 45).

(13, 14, 24, 25) = (33, 34, 44, 45).

(13, 23, 24, 25) = (33, 43, 44, 45).

4-simplices 1: (00, 10, 20, 21, 22).

2: (00, 10, 11, 21, 22).

3: (00, 10, 11, 12, 22).

4: (00, 01, 11, 21, 22).

5: (00, 01, 11, 12, 22).

Approach to TC of Klein bottle

7

6: (00, 01, 02, 12, 22).

7: (30, 40, 50, 51, 52).

8: (30, 40, 41, 51, 52).

9: (30, 40, 41, 42, 52).

10: (30, 31, 41, 51, 52).

11: (30, 31, 41, 42, 52).

12: (30, 31, 32, 42, 52).

13: (03, 13, 23, 24, 25).

14: (03, 13, 14, 24, 25).

15: (03, 13, 14, 15, 25).

16: (03, 04, 14, 24, 25).

17: (03, 04, 14, 15, 25).

18: (03, 04, 05, 15, 25).

19: (33, 43, 53, 54, 55).

20: (33, 43, 44, 54, 55).

21: (33, 43, 44, 45, 55).

22: (33, 34, 44, 54, 55).

23: (33, 34, 44, 45, 55).

24: (33, 34, 35, 45, 55).

3

H 4 (K × K) with local coefficients

We will need to show that a certain class is nonzero in H 4 (K × K; G) with coefficients in a certain local coefficient system G. In this section, we describe the relations in H 4 (K × K; G) for an arbitrary free abelian local coefficient system G. For a ∆-complex X with a single vertex x0 , such as the one just described for K × K, a local coefficient system G is an abelian group G together with an action of π1 (X; x0 ) on G, giving G the structure of Z[π1 (X; x0 )]-module. If Ck denotes the free abelian group generated by the k-cells of X, homomorphisms δk−1 : Hom(Ck−1 , G) → Hom(Ck , G) are defined by (1)

δk−1 (φ)( vi0 , . . . , vik ) = ρi0 ,i1 · φ( vi1 , . . . , vik ) +

k

i=1

(−1)i φ( vi0 , . . . , v i , . . . , vik ),

8

Donald M. Davis

where v i denotes omission of that vertex, and Ď i0 ,i1 is the element of Ď&#x20AC;1 (X; x0 ) corresponding to the edge from vi0 to vi1 . Then H k (X; G) = ker(δk )/ im(δkâ&#x2C6;&#x2019;1 ).

This description is given in [3, p.501].

We have Ď&#x20AC;1 (K; v) = a, b, c /(c = abâ&#x2C6;&#x2019;1 = ba) and Ď&#x20AC;1 (K Ă&#x2014; K; (v, v)) = a, b, c, a , b , c /(c = abâ&#x2C6;&#x2019;1 = ba, c = a b â&#x2C6;&#x2019;1 = b a ), where the primes correspond to the second factor. We will prove the following key result, in which Îł1 , . . . , Îł24 denote the generators corresponding to the 4-cells of K Ă&#x2014;K listed at the end of the previous section, and G is any free abelian local coefficient system.

Theorem 3.1. Let j =

â&#x2C6;&#x2019;1 1

j â&#x2030;Ą 2 mod 3 j â&#x2030;Ą 0, 1 mod 3,

and let Ď&#x2C6; â&#x2C6;&#x2C6; Hom(C4 , G). Then [Ď&#x2C6;] = 0 â&#x2C6;&#x2C6; H 4 (K Ă&#x2014; K; G) if and only if 24

j Ď&#x2C6;(Îłj )

j=1

is 0 in the quotient of G modulo the action of b â&#x2C6;&#x2019; 1, b â&#x2C6;&#x2019; 1, c + 1, and c + 1.

Proof. We find the image of δ : Hom(C3 , G) â&#x2020;&#x2019; Hom(C4 , G) by rowreducing, using only integer operations, the 60-by-24 matrix M whose entries mi,j â&#x2C6;&#x2C6; Z[Ď&#x20AC;1 (K Ă&#x2014; K)] satisfy (2)

δ(Ď&#x2020;)(Îłj ) =

60

mi,j Ď&#x2020;(βi ),

j = 1, . . . , 24.

i=1

Here βi denotes the generator of C3 corresponding to the ith 3-cell.

We list the matrix M below. Each row has two nonzero entries, as each 3-cell is a face of two 4-cells, while each column has five nonzero entries, as each 4-cell is bounded by five 3-cells. For example, m42,1 = c

9

Approach to TC of Klein bottle

because (10, 20, 21, 22) is obtained from (00, 10, 20, 21, 22) by omission of the initial vertex, and ρ00,10 = c.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 −1 0 0 0 0 0 0 0 0 0 0 −1 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 0 0 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 c 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 0 0 0 −1 −1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 c 1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 b 0 0 0 1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 0 0 0 0 0 0 0 0 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 0 0 0 0 0 −1 −1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0

12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 0 0 0 0 0 0 −1 0 0 c 0 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 c

14 0 0 0 0 0 0 0 0 0 0 0 0 1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 c 0 0

16 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −1 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 −1 0 0 0 0 0

19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 0 0 0 −1 0 0 b 0 0 0

20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 b 0 0 0 0

21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 1

22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 1 0 1 0 0 0 0 0 0 0

23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 1 0

24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 b 0 0 0 −1 0 0 0 0 0 1 0 0

Matrix M

The row reduction of this matrix can be done by hand in less than 30 minutes. We find that the row-reduced form has 27 nonzero rows, with its only nonzero elements in rows i = 1, . . . , 23 being 1 in position

10

Donald M. Davis

(i, i) and − i in position (i, 24), while in rows 24 through 27 the only nonzero element is in column 24, and equals b − 1, b − 1, c + 1, and c + 1, respectively.

A row (r1 , . . . , r24 ), with rj = ψ(γj ) for ψ ∈ Hom(C4 , G), is equivalent, modulo the first 23 rows just described, to a row with 0’s in the first 23 positions, and 24 j=1 j rj in the 24th column. The last four rows of the reduced matrix yield the claim of Theorem 3.1.

4

Our specific obstruction class

In [2], the following theorem is proved.

Theorem 4.1. Let X be an n-dimensional ∆-complex with a single vertex x0 , and let π = π1 (X, x0 ). Let I ⊂ Z[π] denote the augmentation ideal. The action of π × π on I by (g, h) · α = gαh−1 makes I a Z[π × π]module, defining a local coefficient system I over X × X. If C1 (X × X) denotes the free abelian group on the set of edges of X × X, then the homomorphism f : C1 (X × X) → I defined by (e1 , e2 ) → [e1 ][e2 ]−1 − 1 defines an element ν ∈ H 1 (X × X; I). Then TC(X) < 2n if and only if ν 2n = 0 ∈ H 2n (X × X; I ⊗2n ), where π × π acts diagonally on I ⊗2n .

The discussion of the function f of this theorem in [2] refers to [9, Ch.6:Thm 3.3], and we use the proof of that result for our interpretation of the function.

When X = K is the Klein bottle, we have π = π1 (K) = {am bn } with the multiplication of these elements determined by the relation ab−1 = ba. Also relevant for us is the element c = ab−1 = ba. The ideal I for us is the free abelian group with basis {αm,n = am bn − 1 : (m, n) ∈ Z × Z − {(0, 0)}}.

Using the numbering of the 1-cells ei of K × K given in Section 2, we obtain that the function f is given as in the following table.

(3)

1 2 3 4 5 6 7 8 i f (ei ) α1,−1 α1,0 α0,1 α−1,−1 α−1,0 α0,−1 0 α0,1 i 9 10 11 12 13 14 15 f (ei ) α1,−2 α0,−1 0 α1,−1 α−1,−2 α−1,−1 0

For example, f (e1 ) = α1,−1 because c = ab−1 , while f (e12 ) = α1,−1 because the edge from 0 to 2 is a, while that from 1 to 2 is b.

Approach to TC of Klein bottle

11

In [3, p.500], it is noted that the Alexander-Whitney formula for cup products in simplicial complexes applies also to ∆-complexes. Our mistake was to overlook the twisting in this formula due to local coefficients.

The correct formula, from [8], is

(4)

(f p ∪ g n−p ) v0 , . . . , vn

= (−1)p(n−p) f (v0 , . . . , vp ) ⊗ ρv0 ,vp g(vp , . . . , vn ),

where ρi,j is as in (1) and is what we overlooked.

We apply this to f 4 (γj ), where f 4 = f ∪ f ∪ f ∪ f with f the function on 1-cells defined above, and γj is any of the 24 4-cells listed in Section 2. For example, f 4 (γ1 ) = f (00, 10) ⊗ cf (10, 20) ⊗ af (20, 21) ⊗ af (21, 22)c−1 = f (e1 ) ⊗ cf (e3 ) ⊗ af (e4 ) ⊗ af (e6 )c−1

(5)

= (ab−1 − 1) ⊗ c(b − 1) ⊗ a(a−1 b−1 − 1) ⊗ a(b−1 − 1)c−1 = (ab−1 − 1) ⊗ (a − ab−1 ) ⊗ (b−1 − a) ⊗ (1 − b−1 ).

Formula (5) equals (T19) in the expansion of the obstruction class in Section 3.2 of [1], after adjusting for different notation. They use x, y for our a, b, and they write their classes as y n xm rather than our am bn .

The relations in π1 (K) must be used to compare these.

For all of our 4-cells, consecutive vertices are constant in one factor, and so only f (ei ) for i ≤ 6 are relevant for f 4 . Once we knew about incorporating the twisting in (4), our obstruction class exactly agreed with the class in Section 3.2 of [1]. Prior to this realization, we worked with a different obstruction class for several months.

5 Approaches to proving that our class is nonzero As just noted, once (4) was understood, our obstruction class agreed with the obstruction class of [1]. They successfully showed that this was nonzero in H 4 (K × K; I ⊗4 ). From our viewpoint, the relations were of the form (1) bαm1 ,n1 ⊗ bαm2 ,n2 ⊗ bαm3 ,n3 ⊗ bαm4 ,n4

−αm1 ,n1 ⊗ αm2 ,n2 ⊗ αm3 ,n3 ⊗ αm4 ,n4 ,

(2) αm1 ,n1 b−1 ⊗ αm2 ,n2 b−1 ⊗ αm3 ,n3 b−1 ⊗ αm4 ,n4 b−1 −αm1 ,n1 ⊗ αm2 ,n2 ⊗ αm3 ,n3 ⊗ αm4 ,n4 ,

12

Donald M. Davis

(3) cαm1 ,n1 ⊗ cαm2 ,n2 ⊗ cαm3 ,n3 ⊗ cαm4 ,n4

+αm1 ,n1 ⊗ αm2 ,n2 ⊗ αm3 ,n3 ⊗ αm4 ,n4 ,

(4) αm1 ,n1 c−1 ⊗ αm2 ,n2 c−1 ⊗ αm3 ,n3 c−1 ⊗ αm4 ,n4 c−1 +αm1 ,n1 ⊗ αm2 ,n2 ⊗ αm3 ,n3 ⊗ αm4 ,n4 ,

and we hoped to show our class could not be reduced to 0 mod these relations.

The factors bαm,n , αm,n b−1 , cαm,n , and αm,n c−1 appearing above are, respectively, 1 : αm,n+(−1)m − α0,1 ,

2 : αm,n−1 − α0,−1 ,

3 : αm+1,n+(−1)m+1 − α1,−1 ,

4 : αm−1,−n−1 − α−1,−1 .

The relations are much simpler in Z[π]⊗4 , and we had a nice form for those. For example, bam1 bn1 ⊗ · · · ⊗ bam4 bn4 ∼ am1 bn1 ⊗ · · · ⊗ am4 bn4 is much simpler than (1). Prior to our understanding of the twisting in (4), we had been able to show our class was not 0 mod the Z[π]⊗4 relations. Once we were made aware of the twisting, we showed our corrected class was 0 mod these relations. Then it was pointed out to us (by Cohen and Vandembroucq) that [2, Lemma 5] shows that the class ν described in our Theorem 4.1 goes to 0 in H 1 (X; Z[π]), and hence our obstruction class ν 4 is zero with coefficients in Z[π]⊗4 . Our methods lent no quick insight toward showing it nonzero with coefficients I ⊗4 .

We wish to thank the authors of [1] for their help in our understanding of this project. Still, we feel that our approach to obtaining the obstruction class, quite different from theirs, is worth publicizing.

Donald M. Davis Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA dmd1@lehigh.edu

References [1] D.C.Cohen and L.Vandembroucq, Topological complexity of the Klein bottle, http://arxiv.org/abs/1612.03133.

Approach to TC of Klein bottle

13

[2] A.Costa and M.Farber, Motion planning in spaces with small fundamental groups, Commun Contemp Math 12 (2010) 107â&#x20AC;&#x201C;119.

[3] S.Eilenberg and J.A.Zilber, Semi-simplicial complexes and singular homology, Annals of Math 51 (1950) 499-513.

[4]

, On products of complexes, Amer Jour Math 75 (1953) 200-204.

[5] M.Farber, Invitation to topological robotics, European Math Society (2008).

[6] A.Hatcher, Algebraic Topology, Cambridge Univ Press (2002).

[7] J.-C.Hausmann, Geometric descriptions of polygon and chain spaces, Contemp Math Amer Math Soc 438 (2007) 47-57.

[8] N.E.Steenrod, Homology with local coefficients, Annals of Math 44 (1943) 610627.

[9] G.W.Whitehead, Elements of Homotopy Theory, Springer-Verlag (1978).

Morfismos, Vol. 21, No. 1, 2017, pp. 15–43 Morfismos, Vol. 21, No. 1, 2017, pp. 15–43

The stability theorem of persistent homology The stability theorem of persistent homology 1 Adam Gardner Adam Gardner

1

Abstract Persistence modules – an important tool for understanding geAbstract ometric properties of data and the central objects of study in persistent homologymodules – are collections of vector indexed by gePersistence – an important toolspaces for understanding the real numbers together maps satisfying certain ba- in ometric properties of with data linear and the central objects of study sic properties.

Persistence modules appear naturally as families persistent homology – are collections of vector spaces indexed by of homology groups associated to filtrations of topological the real numbers together with linear maps satisfying spaces. certain baTwosic equivalent waysPersistence to represent a persistence areasbyfamilies its properties. modules appear module naturally persistence diagram – a multiset of points in the Euclidean plane – of homology groups associated to filtrations of topological spaces. and Two by itsequivalent barcode –ways a multiset of real intervals.

Under certain to represent a persistence module areasby its sumptions (commonly in practice), persistence diagramsatisfied – a multiset of pointsthese in therepresentations Euclidean plane – existand andbyare of the of main in the theory of asits unique. barcode –One a multiset realresults intervals.

Under certain persistence is the Stability Theorem, asserts thatrepresentations small persumptions (commonly satisfied inwhich practice), these turbations of aare persistence in small perturbations of of exist and unique. module One of result the main results in the theory its persistence diagram and barcode.

In this paper, we review the persistence is the Stability Theorem, which asserts that small perevolution of this withmodule emphasis on the results appearing of turbations oftheorem, a persistence result in small perturbations in The Structure and Stability of Persistence Modules (Chazal et the its persistence diagram and barcode. In this paper, we review al., evolution 2012) andofInduced Matchings and the Algebraic Stability of this theorem, with emphasis on the results appearing Persistence Barcodes (Bauer and Lesnick, 2015). in The Structure and Stability of Persistence Modules (Chazal et al., 2012) and Induced Matchings and the Algebraic Stability of 2010 Mathematics Classification: 55N35 2015).

PersistenceSubject Barcodes (Bauer and Lesnick,

Keywords and phrases: Data Analysis, Topology, Persistent Homology 2010 Mathematics Subject Classification: 55N35 Keywords and phrases: Data Analysis, Topology, Persistent Homology

Contents

1 Introduction Contents

16

1

This article corresponds to a project supervised by professors Maia Fraser and Introduction Yael1Karshon and submitted by the author in partial fulfillment of the requirements 16 for the 1title Master of Scienceto- aMathematics, grantedbyinprofessors October Maia 2015 by the and This ofarticle corresponds project supervised Fraser University of Toronto.

Yael Karshon and submitted by the author in partial fulfillment of the requirements for the title of Master of Science - Mathematics, granted in October 2015 by the University of Toronto.

15

15

16

Adam Gardner

2 The 2.1 2.2 2.3

Structure and Stability of Persistence Modules Decomposable Persistence Modules . . . . . . . . . . . .

Interleaving . . . . . . . . . . . . . . . . . . . . . . . . .

Decorated Real Numbers and Persistence Diagrams . . .

3 Induced Matchings and the Algebraic Stability of Persistence Barcodes 3.1 Barcodes . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Dual Modules . . . . . . . . . . . . . . . . . . . . . . . .

3.3 The Structure of Persistence Submodules and Quotients 3.4 Induced Matchings of Barcodes . . . . . . . . . . . . . .

3.5 An Explicit Formulation of the Algebraic Stability Theorem

1

18 20 23 26

30 31 32 34 37 40

Introduction

Persistent homology, a topological tool for analyzing the global, nonlinear, geometric features of data, is the primary object of study in topological data analysis. Frequently in science and engineering, data can be naturally represented as a filtration: namely a collection of topological spaces Xt with t ∈ R such that Xs ⊂ Xt whenever s ≤ t, of which the sublevel set filtration Xt := f −1 ((−∞, t]) for a continous function f : X → R is a common example. To gain insight into the structure of such data, one can apply the homology functor F which takes a topological space X to the vector space Hn (X, k) (with n a fixed nonnegative integer and k a fixed field), and continuous maps f : X → Y , to the induced linear maps f∗ : Hn (X, k) → Hn (Y, k). By the functoriality of F , this yields a collection Mt := Hn (Xt , k) of vector spaces indexed by R together with linear maps ϕM (s, t) := ι(s, t)∗ : Ms → Mt , s ≤ t induced by the inclusion maps ι(s, t) : Xs → Xt such that (i) ϕM (s, t) ◦ ϕM (r, s) = ϕM (r, t) whenever r ≤ s ≤ t, and (ii) ϕM (t, t) =

Mt ,

where Mt : Mt → Mt denotes the identity map. A collection of vector spaces Mt , t ∈ R with linear maps ϕM (s, t), s ≤ t satisfying (i) and (ii) is called a persistence module.

In [8], Edelsbrunner, Letscher and Zomorodian introduce the persis− tence diagram dgm(M ) – a multiset of points in the Euclidean plane

The Stability Theorem of Persistent Homology

17

which encodes information about any given persistence module M satisfying certain finiteness (or “tameness”) conditions – and provide a fast algorithm for computing the persistence diagram when the underlying topological spaces are finite simplicial complexes in R3 . In [3], Carlsson, Collins, Guibas and Zomorodian introduce the barcode BM – a multiset of intervals (“bars”) which encodes nearly identical (but slightly finer) information.

When analyzing real-world data, uncertainties in the data provided are bound to appear, whether this is due to measurement errors, discretization errors or other sources. Therefore, it is essential to distinguish inherent topological features of the data from noise. One way to quantify these errors is via δ−matchings – a bijection between two multisets in the Euclidean plane such that the distance between two corresponding points is at most δ > 0. The bottleneck distance dB (·, ·) is the pseudometric on persistence diagrams defined to be the infimum over all δ > 0 such that there exists a δ-matching of the persistence diagrams. A pseudometric dB (·, ·) can be defined similarly for barcodes.

In [6], Cohen-Steiner, Edelsbrunner and Harer study sublevel-set filtrations X f of real-valued continuous functions f : X → R on topological spaces. They show that, under certain mild finiteness (or “tameness”) assumptions, small perturbations of functions produce small perturbations in the persistence diagram of the persistence modules with Mtf = Hn (Xtf , k); specifically, the bottleneck distance between the persistence diagrams of two functions f and g is bounded above by their

distance in the infinity norm: dB dgm(M f ), dgm(M g ) ≤ ||f − g||∞ .

This is the first incarnation of the Stability Theorem for persistence modules.

A δ−interleaving is an approximate isomorphism of persistence modules, with the error in the approximation quantified by δ. The interleaving distance dI (·, ·) is the pseudometric on persistence modules defined to be the infimum over all δ > 0 such that there exists a δinterleaving of the persistence modules. By replacing the ∞-norm by the interleaving distance, Chazal, Steiner, Glisse, Guibas and Oudot show in [4] that the Stability Theorem of [6] holds for persistence modules M and N satisfying similar tameness conditions: dB (dgm(M ), dgm(N )) ≤ dI (M, N ). This version of stability is called the Algebraic Stability Theorem, as it is a purely algebraic statement. This inequality is in fact an equality – dB (dgm(M ), dgm(N )) = dI (M, N ) – a result known as the Isometry Theorem, which is first proven by Lesnick for so-called pointwise finite-dimensional (PFD) persistence modules in a

18

Adam Gardner

2011 version of [9]. The inequality dB (dgm(M ), dgm(N )) ≥ dI (M, N ) is called the Converse Stability Theorem.

In The Structure and Stability of Persistence Modules [5], Chazal, de Silva, Glisse, and Oudot provide a comprehensive overview of the theory of persistence. Furthermore, they introduce a more general tameness condition, called q-tameness, and prove the Algebraic Stability Theorem and the Converse Stability Theorem for q-tame persistence modules using rectangle measures, functions defined on rectangles in the Euclidean plane with properties analogous to measures.

In Induced Matchings and the Algebraic Stability of Persistence Barcodes [2], Bauer and Lesnick prove a stronger, explicit version of the Algebraic Stability Theorem: given a δ-interleaving of persistence modules of finite dimensional vector spaces, they provide an explicit δ-matching of the respective barcodes.

The current paper will survey these last two papers, focusing on the background and results necessary to prove the Algebraic Stability Theorem (Theorem 3.5.2) and the Converse Stability Theorem (Theorem 2.3.14).

2

The Structure and Stability of Persistence Modules

The following section draws primarily from the exposition of Chazal, de Silva, Glisse and Oudot in The Structure and Stability of Persistence Modules[5]. Some of the notation comes from Bauer and Lesnick[2], and several theorems and definitions first appear in earlier works such as Cohen-Steiner et al.[8] (in which the definition of the persistence diagram first appears).

All vector spaces shall be over a fixed field k.

Definition 2.0.1. Recall from the introduction that a persistence module M is a collection of vector spaces Mt for t ∈ R together with a collection of linear maps ϕM (s, t) : Ms → Mt for every s ≤ t – called transition maps – satisfying the composition law ϕM (s, t) ◦ ϕM (r, s) = ϕM (r, t) whenever r ≤ s ≤ t and such that ϕM (t, t) : Mt → Mt is the identity for every t ∈ R. M is said to be pointwise finite-dimensional (PFD) if each vector space Mt is finite-dimensional.

The Stability Theorem of Persistent Homology

19

Let R = (R, ≤) be the category with objects t ∈ R and a unique morphism s → t whenever s ≤ t. This leads to an equivalent definition of a persistence module: Definition 2.0.2. A persistence module M is a functor M : R → Vect from the category R = (R, ≤) to the category Vect of vector spaces over k. A PFD persistence module M over R is a functor M : R → vect from the category R = (R, ≤) to the category vect of finite-dimensional vector spaces over k.

Definition 2.0.3. A morphism f : M → N between two persistence modules M and N is a collection of linear maps ft : Mt → Nt ,

t∈R

such that the following diagram commutes whenever s ≤ t: Ms

ϕM (s,t)

fs

Ns

Mt ft

ϕN (s,t)

Nt

The composition of morphisms f : M → N and g : N → P is the pointwise composition (g ◦ f )t = gt ◦ ft . This is clearly associative with identity morphism M : M → M equal pointwise to the identity ( M )t = Mt : Mt → Mt .

Remark 2.0.4. In categorical terminology, a morphism f : M → N is a natural transformation between the functors M, N : R → Vect.

Remark 2.0.5. The collection of all persistence models together with morphisms as defined above forms a category, denoted by VectR . This category contains kernels, images and direct sums (categorical coproducts). We construct the direct sum of persistence modules below.

Definition 2.0.6. Let M and N be persistence modules. The direct sum of M and N , denoted by P = M ⊕ N , is the persistence module with vector spaces Pt = Mt ⊕ Nt and linear maps

ϕP (s, t) = ϕM (s, t) ⊕ ϕN (s, t)

from Ps to Pt whenever s ≤ t.

For a collection of persistence modules {Pk | k ∈ K}, the direct sum Pk is defined analogously.

P = k∈K

20

Adam Gardner

Remark 2.0.7. The reader may check that projection maps πM : M ⊕ N → M and πN : M ⊕ N → N , defined pointwise by (πM )t (v ⊕ w) = v and (πN )t (v ⊕ w) = w for v ∈ Mt , w ∈ Nt , are morphisms of persistence modules. Similarly, the pointwise inclusions ιM : M → M ⊕ N and ιN : N → M ⊕ N are morphisms (in fact these are the canonical injections from the definition of a categorical coproduct).

2.1

Decomposable Persistence Modules

We say that I ⊂ R is an interval if I is non-empty and r, t ∈ I implies s ∈ I whenever r ≤ s ≤ t.

Definition 2.1.1. Let I ⊂ R be an interval. The interval module corresponding to I, denoted by C(I), is the persistence module with vector spaces

k if t ∈ I C(I)t = 0 otherwise,

and transition maps ϕC(I) (s, t) =

k

0

if s, t ∈ I otherwise

from C(I)s to C(I)t whenever s ≤ t. Given a persistence module M , we will refer to a submodule N ⊂ M which is isomorphic to C(I) as an interval submodule of M , or more specifically an I-submodule of M .

A persistence module M is said to be decomposable if M is isomorphic to a direct sum of interval modules:

M∼ C(Ik ) = k∈K

where each Ik ⊂ R is an interval.

In general, there may be repeated intervals in a direct sum decomposition, i.e. intervals Ik = Ik with k = k . It is therefore convenient to introduce the notion of a multiset: Definition 2.1.2. A multiset is a pair A = (S, m) where S is a set and m : S → N is a multiplicity function from S to the positive integers N. Intuitively, m(s) is the number of copies of s ∈ S appearing in the multiset A. The representation Rep(A) of the multiset A is the set Rep(A) = {(s, n) | s ∈ S, n ∈ N, n ≤ m(s), }.

The Stability Theorem of Persistent Homology

21

Remark 2.1.3. More generally, we may allow m : S → Card to take values in the proper class of cardinal numbers.

Throughout this paper, we often work with a collection of elements Ik of some set indexed by k ∈ K (for example, see the decomposable module M in Definition 2.1.1). Let A = (S, m) be the multiset with S = {Ik | k ∈ K} and multiplicity function

m(I) = {k ∈ K | Ik = I} ,

I ∈ S.

In this case, we shall use double curly brackets {{Ik | k ∈ K}} := Rep(A) to denote the representation of A. Given a decomposition of a persistence module into a direct sum of interval modules, it is natural to ask if this decomposition is unique.

Theorem 2.1.4 answers this question affirmatively: Theorem 2.1.4 (Unique Decomposition Theorem). Let

M= P Ik = Q Jl k∈K

l∈L

where the P Ik , k ∈ K and QJl , l ∈ L are respectively Ik - and Jl submodules of M . Then {{Ik | k ∈ K}} = {{Jl | l ∈ L}}.

As observed in Chazal et al.[5](Theorem 1.3), this is a corollary of Azumaya[1](Theorem 1). We present another proof below.

Let s ∈ R and let I ⊂ R be an interval. We say that s > I if s > t for every t ∈ I, and s < I if s < t for every t ∈ I. Lemma 2.1.5. Let I, J ⊂ R be intervals. (i) Every morphism f : C(I) → C(J) is of the form

a · k if t ∈ I ∩ J ft = 0 otherwise for some a ∈ k.

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Adam Gardner

(ii) If there exist nonzero morphisms f : C(I) → C(J) and g : C(J) → C(I), then I = J. Proof. If t ∈ I ∩ J then either C(I)t = 0 or C(J)t = 0, so clearly ft = 0.

If t ∈ I ∩ J, then since the only linear maps k → k are multiplication by a scalar, ft = at · k for some at ∈ k. If s, t ∈ I ∩ J with s ≤ t then as = at = a by the commutativity of the following diagram: k

C(I)s as ·

C(I)t at ·

k k

C(J)s

k

C(J)t

This proves Part (i).

For Part (ii), use Part (i) to write

a · k if t ∈ I ∩ J ft = 0 otherwise and gt =

b· 0

k

if t ∈ I ∩ J otherwise.

Suppose I = J, and suppose without loss of generality that there exists some t ∈ I\J. Then either t > J or t < J. If t > J then s < t for every s ∈ I ∩ J. By the commutativity of the diagram k

C(I)s a·

b·

C(I)s

0

k

C(J)s

C(I)t

0

C(J)t = 0 0

k k

C(I)t

we must have ab = 0, which is equivalent to f = 0 or g = 0. The case when t < J is similar.

Proof of the Unique Decomposition Theorem. Fix any interval K of R and consider the submodules P and Q which are direct sums respectively of all P Ik such that Ik = K, or all QJl such that Jl = K. It suffices to show that dim(P )t = dim(Q)t for all t ∈ R.

The Stability Theorem of Persistent Homology

23

Let V = ⊕k∈K,Ik =K P Ik so M = V ⊕ P , and similarly define W so M = W ⊕ Q. Let πV , πP , πW and πQ be projections onto the direct summands V , P , W and Q respectively. While it is not in general true that P=Q, we will show that projection of either P or Q into the other is a monomorphism. The identity on M can be decomposed as M = πW + πQ , and so P = πP |P = πP ◦ (πW + πQ )|P . Let g := πP ◦ πW . We claim P ⊂ ker g so P = πP ◦ πQ |P . Indeed, πW can be further decomposed as a sum of projection morphisms into each of the interval submodules QJl , l ∈ L, Jl = K, and so for any Ik = K, g|P Ik is a sum of morphisms from the K-submodule P Ik to itself, each factoring through a Jl -interval submodule, l ∈ L, Jl = K; by Lemma 10(ii) such morphisms are trivial. Thus, P = πP ◦ πQ |P and it follows that dim Pt = dim πQ (P )t ≤ dim Qt , ∀t ∈ R. The analogous argument with roles of P and Q reversed gives dim Qt ≤ dim Pt .

Remark 2.1.6. Theorem 2.1.4 does not guarantee that all persistence modules are decomposable, but merely that if a decomposition exists, it is unique. However, the following theorem guarantees that a large class of persistence modules are decomposable: Theorem 2.1.7. Every pointwise finite-dimensional (PFD) persistence module is decomposable.

The proof, which is beyond the scope of this paper, appears in [7] (Theorem 1.1).

2.2

Interleaving

Let M and N be persistence modules, and let δ ≥ 0. Following the notation of Bauer and Lesnick in [2], we define M (δ) to be the persistence module with vector spaces M (δ)t = Mt+δ and transition maps ϕM (δ) (s, t) = ϕM (s + δ, t + δ) for s ≤ t. Given a morphism f : M → N , we define the morphism f (δ) : M (δ) → N (δ) by f (δ)t = ft+δ .

Definition 2.2.1. The δ−shift functor (δ)(·) : VectR → VectR is the functor sending a persistence module M to M (δ) and a morphism f : M → N to f (δ) : M (δ) → N (δ).

Remark 2.2.2. For δ > 0 and a, b ∈ R with a < b, we have C [a, b] (δ) =

C(Ik ) then M (δ) ∼ C [a − δ, b − δ] . More generally, if M ∼ = = k∈K

C(Ik (δ)), where Ik (δ) is the interval Ik with its endpoints shifted to k∈K

the left by δ.

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Adam Gardner

Definition 2.2.3. Let M be a persistence module. For any real number ≥ 0, the transition maps of M define a canonical morphism ϕ M : M → M ( ) called the − transition morphism: (ϕ M )t = ϕM (t, t + ) for every t ∈ R. Remark 2.2.4. ϕ M is a morphism because the diagram ϕM (s,t)

Ms

Mt

ϕM (s,s+ )

Ms+

ϕM (t,t+ )

ϕM (s+ ,t+ )

Mt+

commutes whenever s ≤ t by the definition of a persistence module.

Definition 2.2.5. Two persistence modules M and N are said to be δ −interleaved if there are morphisms f : M → N (δ) and g : N → M (δ) 2δ such that g(δ) ◦ f = ϕ2δ M and f (δ) ◦ g = ϕN . More explicitly, we require that the diagrams Ms

ϕM (s,t)

Mt

fs

Ns+δ Ns gs

Ms−δ

fs−δ

ft ϕN (s+δ,t+δ)

Nt+δ

ϕN (s,t)

Nt gt

ϕM (s+δ,t+δ) Ms+δ Mt+δ

ϕM (s−δ,s+δ)

Ms+δ

gs

Ns Ns−δ

ϕN (s−δ,s+δ) gs−δ

Ns+δ

fs

Ms

commute whenever s < t. f and g are then called δ−interleavings, and f is said to be a δ−inverse of g (and vice-versa).

Remark 2.2.6. A δ-inverse g is not unique in general. For example, let M and N be any two persistence module such that ϕM (t, t + 2δ) = 0 : Mt → Mt+2δ and ϕN (t, t + 2δ) = 0 : Nt → Nt+2δ for all t ∈ R – for instance, M = C([0, δ)) and N = C([p, p + δ0 ]) with p ∈ R and 0 < δ0 < δ. Then M and N are δ-interleaved, where the δ-interleaving f in Definition 2.2.5 can be taken to be the zero morphism f = 0 : M → N (δ). We note that any morphism g : N → M (δ) is a δ-inverse of f .

The Stability Theorem of Persistent Homology

25

Remark 2.2.7. If two persistence modules M and N are δ-interleaved, then they are -interleaved for every ≥ δ. Indeed, if f : M → N (δ) is a δ-interleaving with δ-inverse g : N → M (δ), then the composition −δ f ( − δ) ◦ ϕ −δ M is an -interleaving with -inverse g( − δ) ◦ ϕN , which −δ can be easily seen by observing that f ( − δ) ◦ ϕ −δ M = ϕN (δ) ◦ f and −δ g( − δ) ◦ ϕ −δ N = ϕM (δ) ◦ g.

Remark 2.2.8. Let M , N and P be persistence modules. If M and N are δ1 -interleaved, and N and P are δ2 -interleaved, then M and P are (δ1 +δ2 )-interleaved. Indeed, if f1 : M → N (δ1 ) is a δ1 -interleaving with δ1 -inverse g1 : N → M (δ1 ), and if f2 : N → P (δ2 ) is a δ2 -interleaving with δ2 -inverse g2 : P → N (δ2 ), then the composition f = f2 (δ1 ) ◦ f1 : M → P (δ1 + δ2 ) is a (δ1 + δ2 )-interleaving with (δ1 + δ2 )-inverse g = g1 (δ2 ) ◦ g2 : P → M (δ1 + δ2 ).

Definition 2.2.9. The interleaving distance dI (·, ·) between two persistence modules M and N is the infimum over all non-negative real numbers such that M and N are δ-interleaved: dI (M, N ) = inf{δ ≥ 0 | there exists a δ-interleaving between M and N } Lemma 2.2.10. The interleaving distance satisfies the triangle inequality: for any three persistence modules M , N and P , we have dI (M, P ) ≤ dI (M, N ) + dI (N, P ).

Proof. By Remark 2.2.8, if M and N are δ1 -interleaved and N and P are δ2 -interleaved, then M and P are (δ1 + δ2 )-interleaved. The result follows by taking the infimum over all such δ1 and δ2 .

Proposition 2.2.11. Let M =

Mj and let N =

j∈J

same indexing set J ). Then

Nj (with the

j∈J

dI (M, N ) ≤ sup dI (Mj , Nj ). j∈J

Proof. If for every j ∈ J there exists an -interleaving fj : Mj → Nj ( ), then f = j∈J fj is an -interleaving of M and N , and so dI (M, N ) ≤ .

Thus by Remark 2.2.7, any upper bound of the interleaving distances dI (Mj , Nj ) must then be an upper bound of dI (M, N ); in particular, this applies to the supremum of the dI (Mj , Nj ).

26

2.3

Adam Gardner

Decorated Real Numbers and Persistence Diagrams

We now introduce the notion of decorated real numbers, which simplify interval notation and play an important role in the approach of [5] and [2].

Definition 2.3.1. The decorated real numbers, denoted by D, are the collection of ordered pairs (p, Âą) with p â&#x2C6;&#x2C6; R â&#x20AC;&#x201C; which we shall denote henceforth by pÂą , or pâ&#x2C6;&#x2014; if the â&#x20AC;&#x153;decorationâ&#x20AC;? â&#x2C6;&#x2014; â&#x2C6;&#x2C6; {â&#x2C6;&#x2019;, +} is unspecified â&#x20AC;&#x201C; together with Âąâ&#x2C6;&#x17E;. Ordering {â&#x2C6;&#x2019;, +} by setting â&#x2C6;&#x2019; < +, we endow D with the lexicographic order, with â&#x2C6;&#x2019;â&#x2C6;&#x17E; and +â&#x2C6;&#x17E; the minimum and

maximum elements of the order respectively. Explicitly, pâ&#x2C6;&#x2014; < q â&#x2C6;&#x2014; if and only if either p < q, or p = q and â&#x2C6;&#x2014; = â&#x2C6;&#x2019;, â&#x2C6;&#x2014; = +.

Remark 2.3.2. The extended real numbers, denoted by R, are the real numbers R with the standard order together with a maximal element â&#x2C6;&#x17E; and a minimal element â&#x2C6;&#x2019;â&#x2C6;&#x17E; (i.e. R = R â&#x2C6;Ş {â&#x2C6;&#x2019;â&#x2C6;&#x17E;, â&#x2C6;&#x17E;}). The extended real numbers R can be obtained from the decorated real numbers D by â&#x20AC;&#x153;forgettingâ&#x20AC;? the decorations â&#x20AC;&#x201C; namely by identifying p+ and pâ&#x2C6;&#x2019; with p (p â&#x2C6;&#x2C6; R).

Remark 2.3.3. We can identify ordered pairs (pâ&#x2C6;&#x2014; , q â&#x2C6;&#x2014; ) â&#x2C6;&#x2C6; D Ă&#x2014; D with intervals of real numbers whenever pâ&#x2C6;&#x2014; < q â&#x2C6;&#x2014; as in the following table: pâ&#x2C6;&#x2019; p+ â&#x2C6;&#x2019;â&#x2C6;&#x17E;

qâ&#x2C6;&#x2019; [p, q) (p, q) (â&#x2C6;&#x2019;â&#x2C6;&#x17E;, q)

q+ [p, q] (p, q] (â&#x2C6;&#x2019;â&#x2C6;&#x17E;, q]

â&#x2C6;&#x17E; [p, â&#x2C6;&#x17E;) (p, â&#x2C6;&#x17E;) (â&#x2C6;&#x2019;â&#x2C6;&#x17E;, â&#x2C6;&#x17E;)

When we refer to the interval to the ordered pair (pâ&#x2C6;&#x2014; , q â&#x2C6;&#x2014; ), â&#x2C6;&#x2014; corresponding

â&#x2C6;&#x2014; we shall use the notation p , q .

Definition 2.3.4. Let M be a decomposable persistence module with

Mâ&#x2C6;ź C pâ&#x2C6;&#x2014;k , qkâ&#x2C6;&#x2014; . = kâ&#x2C6;&#x2C6;K

The decorated persistence diagram of M is the multiset of pairs of decorated real numbers Dgm(M ) = {{(pâ&#x2C6;&#x2014;k , qkâ&#x2C6;&#x2014; ) | k â&#x2C6;&#x2C6; K}}.

The undecorated persistence diagram of M is the multiset of pairs of (undecorated, extended) real numbers dgm(M ) = {{(pk , qk ) | k â&#x2C6;&#x2C6; K}}.

The Stability Theorem of Persistent Homology

27

Remark 2.3.5. By Theorem 2.1.4, the (un)decorated persistence diagram of a decomposable persistence module does not depend on the choice of decomposition.

Definition 2.3.6. A function σ is said to be a partial matching between sets A and B if σ is a bijection between a subset A ⊂ A and a subset B ⊂ B. In other words, we say that σ is a partial matching between A and B if σ is a bijection with dom(σ) ⊂ A and im(σ) ⊂ B. We write σ : A B to mean “σ is a partial matching between A and B.” We say that a pair of points a ∈ A and b ∈ B are matched if a ∈ dom(σ), b ∈ im(σ) and σ(a) = b.

The composition of a partial matching σ1 between A and B and a partial matching σ2 between B and C is the partial matching σ2 ◦ σ1 = {(a, c) | ∃b ∈ B such that (a, b) ∈ σ1 and (b, c) ∈ σ2 }.

Remark 2.3.7. In [2] (Bauer and Lesnick), the authors refer to partial matchings simply as matchings. We use terminology partial matching which appears in [5] (Chazal et al.) to emphasize that only some of the elements in partially matched sets are matched.

We shall soon define a distance between subsets of the extended plane R2 using partial matchings, but first we must choose a point-topoint distance. Let d∞ (·, ·) denote the infinity norm on the extended plane, defined by

d∞ (p, q), (r, s) = max{|r − p|, |s − q|}.

Here we extend |r − p| to allow for p or r to be infinite by setting |r − p| = ∞ if one of p or r is infinite and p = r, and |r − p| = 0 if p = r – for instance, we have |∞ − 2| = |∞ − (−∞)| = ∞, but |∞ − ∞| = 0.

If (p, q) ∈ R2 and S ⊂ R2 , let

d∞ (p, q), S = inf d∞ (p, q), (r, s) . (r,s)∈S

In particular, letting

∆ = {(p, q) ∈ R2 | q = p}, denote the diagonal, it is easy to see that

1 d∞ (p, q), ∆ = |q − p|.

2

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Adam Gardner

Definition 2.3.8. A partial matching Ï&#x192; between two subsets A and B of R2 is called a δâ&#x2C6;&#x2019;matching if all of the following statements are true: â&#x20AC;¢ if a â&#x2C6;&#x2C6; A and b â&#x2C6;&#x2C6; B are matched in Ï&#x192; (i.e. Ï&#x192;(a) = b) then dâ&#x2C6;&#x17E; (a, b) â&#x2030;¤ δ; â&#x20AC;¢ if a â&#x2C6;&#x2C6; A is unmatched in Ï&#x192; (i.e. a â&#x2C6;&#x2C6; dom(Ï&#x192;)) then dâ&#x2C6;&#x17E; (a, â&#x2C6;&#x2020;) â&#x2030;¤ δ; â&#x20AC;¢ if b â&#x2C6;&#x2C6; B is unmatched in Ï&#x192; (i.e. b â&#x2C6;&#x2C6; im(Ï&#x192;)) then dâ&#x2C6;&#x17E; (b, â&#x2C6;&#x2020;) â&#x2030;¤ δ.

We say that A and B are δâ&#x2C6;&#x2019;matched if there exists a δ-matching between A and B. Remark 2.3.9. Let A, B, C â&#x160;&#x201A; R2 . If A and B are δ1 -matched, and B and C are δ2 -matched, then A and C are (δ1 + δ2 )-matched. Indeed, let Ï&#x192;1 : A B be a δ1 -matching and let Ï&#x192;2 : B C be a δ2 -matching. We verify that Ï&#x192; = Ï&#x192;2 â&#x2014;¦ Ï&#x192;1 : A C is a (δ1 + δ2 )-matching: â&#x20AC;¢ If a â&#x2C6;&#x2C6; A and c â&#x2C6;&#x2C6; C are matched in Ï&#x192;, then there is some b â&#x2C6;&#x2C6; B such that (a, b) â&#x2C6;&#x2C6; Ï&#x192;1 and (b, c) â&#x2C6;&#x2C6; Ï&#x192;2 , and so dâ&#x2C6;&#x17E; (a, c) â&#x2030;¤ dâ&#x2C6;&#x17E; (a, b) + dâ&#x2C6;&#x17E; (b, c) â&#x2030;¤ δ1 + δ2 . â&#x20AC;¢ If a â&#x2C6;&#x2C6; A is unmatched in Ï&#x192;, then either a is unmatched in Ï&#x192;1 , in which case dâ&#x2C6;&#x17E; (a, â&#x2C6;&#x2020;) â&#x2030;¤ δ1 â&#x2030;¤ δ1 + δ2 ; or a is matched in Ï&#x192;1 to some b â&#x2C6;&#x2C6; B which is unmatched in Ï&#x192;2 , in which case dâ&#x2C6;&#x17E; (a, â&#x2C6;&#x2020;) â&#x2030;¤ dâ&#x2C6;&#x17E; (a, b) + dâ&#x2C6;&#x17E; (b, â&#x2C6;&#x2020;) â&#x2030;¤ δ1 + δ2 . â&#x20AC;¢ The case of an unmatched point c â&#x2C6;&#x2C6; C is similar to the previous case.

Definition 2.3.10. The bottleneck distance dB (·, ·) between two subsets A and B of R2 is the infimum over all non-negative real numbers such that A and B are δ-matched: dB (A, B) = inf{δ â&#x2030;¥ 0 | there exists a δ-matching between A and B} Lemma 2.3.11. The bottleneck distance satisfies the triangle inequality: for any three subsets A, B and C of R2 , we have dB (A, C) â&#x2030;¤ dB (A, B) + dB (B, C).

The Stability Theorem of Persistent Homology

29

Proof. By Remark 2.3.9, if A and B are δ1 -matched and B and C are δ2 -matched, then A and C are (δ1 + δ2 )-matched. The result follows by taking the infimum over all such δ1 and δ2 . â&#x2C6;&#x2014; â&#x2C6;&#x2014; â&#x2C6;&#x2014; â&#x2C6;&#x2014; Proposition â&#x2C6;&#x2014;p ,â&#x2C6;&#x2014;q and r , s be intervals, and let M = â&#x2C6;&#x2014; â&#x2C6;&#x2014; 2.3.12. Let and N = C r , s be the corresponding interval modules.

C p ,q Then dI (M, N ) â&#x2030;¤ dâ&#x2C6;&#x17E; ((p, q), (r, s)).

Proof. We must show that for every δ > dâ&#x2C6;&#x17E; ((p, q), (r, s)), M and N are δ-interleaved. We do this by noting that the maps f : M â&#x2020;&#x2019; N (δ) and g : N â&#x2020;&#x2019; M (δ) defined by ft =

gt =

and

k

if t â&#x2C6;&#x2C6; pâ&#x2C6;&#x2014; , q â&#x2C6;&#x2014; â&#x2C6;Š râ&#x2C6;&#x2014; â&#x2C6;&#x2019; δ, sâ&#x2C6;&#x2014; â&#x2C6;&#x2019; δ otherwise,

k

if t â&#x2C6;&#x2C6; pâ&#x2C6;&#x2014; â&#x2C6;&#x2019; δ, q â&#x2C6;&#x2014; â&#x2C6;&#x2019; δ â&#x2C6;Š râ&#x2C6;&#x2014; , sâ&#x2C6;&#x2014; otherwise,

0

0

provide well-defined δ-inverse δ-interleavings.

Proposition 2.3.13. Let pâ&#x2C6;&#x2014; , q â&#x2C6;&#x2014; be an interval, and let M = C pâ&#x2C6;&#x2014; , q â&#x2C6;&#x2014; .

Then 1 dI (M, 0) = |q â&#x2C6;&#x2019; p| = dâ&#x2C6;&#x17E; ((p, q), â&#x2C6;&#x2020;).

2 Proof. The only morphisms to and from the zero module are the zero maps. Therefore a pair of morphisms f : M â&#x2020;&#x2019; 0(δ) = 0 and g : 0 â&#x2020;&#x2019; M (δ) are δ-inverse δ-interleavings if and only if Ď&#x2022;2δ M = g(δ) â&#x2014;Ś f , which = 0.

This last equality holds if 2δ > |q â&#x2C6;&#x2019; p| and fails simplifies to Ď&#x2022;2δ M to hold if 2δ < |q â&#x2C6;&#x2019; p|.

Theorem 2.3.14 (The Converse Stability Theorem for Decomposable Persistence Modules). Let M , N be decomposable persistence modules.

Then dI (M, N ) â&#x2030;¤ dB (dgm(M ), dgm(N )).

This theorem is called the Converse Stability Theorem because the reverse inequality, which was studied first, is true for pointwise finitedimensional (PFD) persistence modules:

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Adam Gardner

Theorem 2.3.15 (The Algebraic Stability Theorem for PFD Persistence Modules). Let M , N be PFD persistence modules. Then dB (dgm(M ), dgm(N )) ≤ dI (M, N ).

We shall prove a stronger version of Algebraic Stability Theorem in Section 3 (see Theorem 3.5.2), following the proof originally appearing in [2] as Theorem 6.4.

Proof of the Converse Stability Theorem. Let δ ≥ 0, and suppose that there is a δ-matching σ of dgm(M ) and dgm(N ). Re-write M and N in the form M=

Mk

k∈K

N=

Nk

k∈K

where we choose Mk , Nk such that either • Mk is an Ik -submodule of M and Nk is a Jk -submodule of N , where Ik and Jk are a pair of matched intervals; • Mk is an Ik -submodule of M where Ik is unmatched, and Nk = 0; or • Nk is a Jk -submodule of N where Jk is unmatched, and Mk = 0.

By Propositions 2.3.12 and 2.3.13, dI (Mk , Nk ) ≤ δ for each of the three possible cases above. By Proposition 2.2.11, dI (M, N ) ≤ δ. The result follows by taking the infimum over all δ ≥ 0 such that there exists a δ-matching of dgm(M ) and dgm(N ).

3

Induced Matchings and the Algebraic Stability of Persistence Barcodes

In this section we summarize the results of Bauer and Lesnick from Induced Matchings and the Algebraic Stability of Persistence Barcodes [2], following their exposition.

The Stability Theorem of Persistent Homology

3.1

31

Barcodes

Definition 3.1.1. A barcode is (a representation of) a multiset of intervals (see Definition 2.1.2).

Definition 3.1.2. Let M be a decomposable persistence module with

Mâ&#x2C6;ź C Ik . = kâ&#x2C6;&#x2C6;K

The barcode of M is the multiset of intervals BM = {{Ik | k â&#x2C6;&#x2C6; K}} Remark 3.1.3. The decorated persistence diagram and the barcode of a decomposable persistence module M contain exactly the same information: in fact, there is a canonical bijection of multisets )â&#x2020;&#x2019; â&#x2C6;&#x2014; â&#x2C6;&#x2014;Dgm(M

â&#x2C6;&#x2014; â&#x2C6;&#x2014; BM which sends (p , q ) â&#x2C6;&#x2C6; Dgm(M ) to the interval p , q .

Remark 3.1.4. By Theorem 2.1.4, the barcode of a decomposable persistence module does not depend on the choice of decomposition.

Remark 3.1.5. Recall that a persistence module M is pointwise finitedimensional (PFD) if Mt is finite- dimensional for every t â&#x2C6;&#x2C6; R (Definition 2.0.1), and that every PFD persistence module is decomposable (Theorem 2.1.7). In particular, to every PFD persistence module M there corresponds a unique barcode BM .

Remark 3.1.6. The barcode of M (δ) is simply the barcode of M shifted to the left by δ (see Definition 2.2.1 and Remark 2.2.2).

Given a barcode D, we define a new barcode D by D = {I â&#x2C6;&#x2C6; D : [t, t + ] â&#x160;&#x201A; I for some t â&#x2C6;&#x2C6; R}.

In words, D is the collection of intervals in D of length strictly greater than together with the closed intervals of length .

Definition 3.1.7. Let C and D be barcodes. A δâ&#x2C6;&#x2019;matching is a partial matching Ď&#x192; : C D such that (i) C 2δ â&#x160;&#x201A; dom(Ď&#x192;), (ii) D2δ â&#x160;&#x201A; im(Ď&#x192;),

32

Adam Gardner

(iii) If σ b, d = b , d then

b, d ⊂ b − δ, d + δ

b , d ⊂ b − δ, d + δ .

The bottleneck distance dB (·, ·) between barcodes C and D is the infimum over all δ ≥ 0 such that C and D are δ-matched: dB (C, D) = inf{δ ≥ 0 | C and D are δ-matched}.

Remark 3.1.8. If M and N are persistence modules, then dB (BM , BN ) = dB (dgm(M ), dgm(N )), which justifies the reuse of the notation dB (·, ·).

Note however that a δ-matching of the barcodes BM and BN is strictly stronger than a δ-matching of persistence diagrams dgm(M ) and dgm(N ), in the sense that if BM and BN are δ-matched then dgm(M ) and dgm(N ) are δ-matched, but the converse is not true in general. Roughly speaking, this is because Definition 2.3.8 does not distinguish between distinct intervals with identical as it “forgets” decorations. endpoints,

For instance, let M = C [p, q] and N = C (p + δ, q − δ) where p < q and 0 ≤ 2δ < q − p. Then dgm(M ) = {(p, q)} ⊂ R2 and dgm(N ) = {(p + δ, q − δ)} ⊂ R2 are δ-matched (simply match the point (p, q) with (p+δ, q−δ)). However, BM = {[p, q]} and BN = {(p+δ, q−δ)} are not δ-matched: indeed, since [p, p + 2δ] ⊂ [p, q], by condition (i) of Definition 3.1.7 [p, q] must be matched to (p+δ, q−δ) by any δ-matching; but since [p, q] ⊂ (p, q), by condition (iii) [p, q] and (p + δ, q − δ) cannot be matched.

3.2

Dual Modules

Let Rop = (R, ≥) be the opposite category of R, that is the category with objects t ∈ R and a unique morphism s → t whenever s ≥ t (instead of whenever s ≤ t). Let Neg : R → Rop denote the functor which sends a real number t to −t and the morphism s ≤ t to −s ≥ −t.

Let (·)∗ : Vect → Vect denote the duality (contravariant) functor.

Given a persistence module M : R → Vect, taking the duals of all vector spaces and transition maps gives a functor M † : Rop → Vect.

Define the dual of M to be the persistence module M ∗ = M † ◦ Neg : R → Vect. Explicitly, M ∗ has vector spaces (M ∗ )t = (M−t )∗ and transition maps ϕM ∗ (s, t) = ϕM (−t, −s)∗ .

Given a morphism of persistence modules f : M → N , we define the dual morphism f ∗ : N ∗ → M ∗ by letting (f ∗ )t = (f−t )∗ .

The Stability Theorem of Persistent Homology

33

With these identifications, dualization is a contravariant endofunctor (·)∗ : VectR → VectR . When M and N are PFD, under the canonical identifications M = M ∗∗ and N = N ∗∗ we have f ∗∗ = f .

For a barcode D, let D∗ = {−I : I ∈ D} where we define −I = {−t : t ∈ I}.

Proposition 3.2.1. If M is a PFD persistence module, then BM ∗ = (BM )∗ .

C(I). It suffices to Proof. Without loss of generality, let M = I∈B M

show that M ∗ ∼ C(−I). We shall prove this in two steps: = (i) If N =

I∈BM

Nk is PFD then N ∗ =

k∈K

k∈K

Nk∗ .

(ii) For any interval I ⊂ R, C(I)∗ ∼ = C(−I).

For (i), let s ≤ t. Since the N−t and N−s , −t ≤ −s are finite dimensional, only finitely many of the terms ϕNk (−t, −s) in ϕN (−t, −s) =

ϕNk (−t, −s) are nonzero. Therefore ϕN (−t, −s)∗ = ϕNk (−t, −s)∗ , k∈K k∈K

ϕNk∗ (s, t) whenever s ≤ t, which proves the result. i.e. ϕN ∗ (s, t) = k∈K

For (ii), observe that

(C(I)∗ )t = (C(I)−t )∗ = Since

∗ k

k∗ , 0,

if t ∈ −I otherwise.

the transition maps of C(I ∗ ) are

k∗ , if s, t ∈ −I ∗ ϕC(I)∗ (s, t) = ϕC(I) (−t, −s) = 0, otherwise.

=

k∗ ,

The isomorphism k → k∗ sending a ∈ k to a · k ∈ k∗ therefore leads to an isomorphism ft : C(−I) → C(I)∗ :

a · k , if t ∈ −I ft (a) = 0, otherwise.

Now applying (i) and (ii) above, we see that

M∗ = C(I)∗ ∼ C(−I), = I∈BM

as required.

I∈BM

34

3.3

Adam Gardner

The Structure of Persistence Submodules and Quotients

A partially ordered set (S, <) is said to be enumerated if there is an order-preserving injection S â&#x2020;&#x2019; N into the positive integers with the standard order. We may write S = {s1 , s2 , ...} with s1 < s2 < ... where this sequence terminates if S is finite.

Given two enumerated sets S = {s1 , s2 , ...} and T = {t1 , t2 , ...} with |S| â&#x2030;¤ |T |, there is a canonical injection ÎąST : S â&#x2020;&#x2019; T defined by ÎąST (si ) = ti .

Remark 3.3.1. If S, T, U are enumerated sets with |S| â&#x2030;¤ |T | â&#x2030;¤ |U |, then the canonical injections trivially satisfy the composition law ÎąSU = ÎąTU â&#x2014;Ś ÎąST .

Let M be a PFD persistence module and let D denote the decorated

real numbers (see Definition 2.3.1). For any b â&#x2C6;&#x2C6; D, define b, ¡ M to be (the of) the multiset of intervals I â&#x2C6;&#x2C6; BM of the form representation

I = b, d for some d â&#x2C6;&#x2C6; D. Clearly

BM = b, ¡ M . bâ&#x2C6;&#x2C6;D

Order b, ¡ M by reverse inclusion, so that larger intervals come before

smaller ones. More precisely, let b, d k denote the k th interval of the

form b, d in the representation of b, ¡ M . We say that b, d k <

b, d k if either d > d , or d = d and k < k .

Remark 3.3.2. b, ¡ M , < is an enumerated set since < defined above is clearly a total order and the number of predecessors of an interval b, d â&#x2C6;&#x2C6; b, ¡ M , < must be finite for M to be PFD.

Dually, define ¡, d M to be the collection of intervals I â&#x2C6;&#x2C6; BM of the

form b, d for some b â&#x2C6;&#x2C6; D, so that

BM = ¡, d M . dâ&#x2C6;&#x2C6;D

Ordering distinct intervals by reverse inclusion (and ordering repeated intervals as above) makes ¡, d M into an enumerated set. Note that in

this case b, d < b , d if b < b .

The Stability Theorem of Persistent Homology

35

Definition 3.3.3. A morphism j : M → N of persistence modules M and N is a monomorphism if jt : Mt → Nt is injective for all t ∈ R. A morphism q : M → N of persistence modules M and N is an epimorphism if qt : Mt → Nt is surjective for all t ∈ R. We shall use the notation j : M → N for monomorphisms and q : M N for epimorphisms. The reader may verify that this definition agrees with the standard categorical definitions of monomorphisms and epimorphisms in the category VectR of persistence modules.

Theorem 3.3.4 (Structure of Persistence Submodules and Quotients).

Let M and N be PFD persistence modules. (i) If there is a monomorphism M → N , then for each d ∈ D

·, d ≤ ·, d M N

and the union of the canonical injections ·, d M → ·, d N , d ∈ D

defines an injection BM → BN sending each interval b, d ∈

BM to an interval b , d with b ≤ b (equivalently, with b , d ≤ b, d ).

(ii) Dually, if there is epimorphism M N , then for each b ∈ D

b, · ≥ b, · M

N

and the union of the canonical injections b, · M ← b, · N , b ∈ D

defines an BM ← BN sending each b, d ∈ BN to an

injection interval b, d with d ≤ d (equivalently, with b, d ≥ b, d ).

Remark 3.3.5. Abusing notation, we shall refer to the injection in part (i) (resp. (ii)) of Theorem 3.3.4 as the canonical injection BM → BN (resp. BM ← BN ) when the conditions of (i) (resp. (ii)) hold.

Informally, the canonical injection in Theorem 3.3.4(i) (resp. (ii)) sends an interval in M (resp. N ) to a larger interval in N (resp. M ) with the same right (resp. left) endpoint. Here “larger” means larger in the sense of length, not in the sense of the partial order we have defined on intervals – for instance, the interval [0, 2] is “larger” than the interval [0, 1], but [0, 2] < [0, 1] in the partial order defined above. Note that these injections are canonical with respect to the representation of BM and BN , not the multisets BM and BN , since permuting identical intervals in BM without a predefined order would yield an equally sensible injection.

36

Adam Gardner

For I = b, d , define ¡, I M to be the multiset

¡, I

M

:= {J = b , d k â&#x2C6;&#x2C6; ¡, d M | b , d â&#x2030;¤ I},

the multiset of predecessors of I in ¡, d M with respect to reverse inclusion.

Lemma 3.3.6. Let I be an interval. If there exists a monomorphism of PFD persistence modules j : M â&#x2020;&#x2019; N then

¡, I â&#x2030;¤ ¡, I .

M N

Proof. Let I = b, d . Without loss of generality, assume M=

C(J)

Jâ&#x2C6;&#x2C6;BM

N=

C(J).

Jâ&#x2C6;&#x2C6;BN

Let U â&#x160;&#x201A; M and V â&#x160;&#x201A; N be the submodules U =

Jâ&#x2C6;&#x2C6; ¡,I

V =

Jâ&#x2C6;&#x2C6; ¡,I

C(J) M

C(J).

N

Given an interval J, we say that t > J if t > s for every s â&#x2C6;&#x2C6; J. Since M and N are PFD, for any s â&#x2C6;&#x2C6; I there must be finitely many intervals some t â&#x2C6;&#x2C6; I such J â&#x2C6;&#x2C6; BM â&#x2C6;Ş BN with s â&#x2C6;&#x2C6; J; it follows that there must be , d â&#x2C6;&#x2C6; B â&#x2030;¤ b and d < d. b â&#x2C6;Ş B with b that t > b , d whenever M N

Clearly dim Ut = ¡, I M and dim Vt = ¡, I N .

We claim that jt (Ut ) â&#x160;&#x201A; Vt . By the choice of t, we have Ut =

sâ&#x2C6;&#x2C6;I sâ&#x2030;¤t

Vt =

sâ&#x2C6;&#x2C6;I sâ&#x2030;¤t

im Ď&#x2022;M (s, t) â&#x2C6;Š im Ď&#x2022;N (s, t) â&#x2C6;Š

ker Ď&#x2022;M (t, r)

r>I

r>I

ker Ď&#x2022;N (t, r).

The Stability Theorem of Persistent Homology

37

For each s ∈ I, we have jt (im ϕM (s, t)) ⊂ im ϕN (s, t) by the commutativity of the diagram Ms

ϕM (s,t)

js

Ns

Mt jt

ϕN (s,t)

Nt

Similarly, jt (ker ϕM (t, r)) ⊂ ker ϕN (t, r) by the commutativity of the diagram Mt

ϕM (t,r)

jt

Nt

Mt jr

ϕN (t,r)

Nr

We conclude that jt (Ut ) ⊂ Vt . Since jt is an injection, we have dim Ut ≤ dim Vt , completing the proof.

Proof of Theorem 3.3.4.

Suppose that exists a monomorphism M →

there N . To show that ·, d M ≤ ·, d N , it suffices to show that if

i ≤ ·, d M then i ≤ ·, d N . Let I = b, d denote the ith inter

val of ·, d M . Then for 1 ≤ j ≤ i, ·, I M contains the j th interval of

·, d M and so

i ≤ ·, I M ≤ ·, I N ≤ ·, d N

where the second from Lemma inequality follows 3.3.6.

th Let I = b , d denote the i interval of ·, d N . Then the canonical

injection BM → BN sends I to I . Since i ≤ ·, I N , we must have

I ∈ ·, I N or equivalently b ≤ b. This completes the proof of part (i).

Part (ii) follows from part (i) by a duality argument. Given an epimorphism of PFD persistence modules q : M N , the dual q ∗ : N ∗ → M ∗ is a monomorphism. By part (i), q ∗ induces an injection ι : BN ∗ → BM ∗ . By Proposition 3.2.1, this in turn induces an injection BN → BM by sending I ∈ BN to −ι(−I), which

is exactly the canonical injection.

By part (i), we see that I = b, d gets sent to I = − −

d , −b = b, d with −d ≤ −d, proving part (ii).

3.4

Induced Matchings of Barcodes

Given a morphism of PFD persistence modules f : M → N , we shall define a partial matching χf : BM BN induced by f .

38

Adam Gardner

We first define this partial matching for monomorphisms and epimorphisms. If j : M → N is a monomorphism, then we define χj : BM BN to be the canonical injection ι : BM → BN of Theorem 3.3.4(i). If q : M N is an epimorphism, then we define χq : BM BN to be the inverse of the canonical injection ι : BN → BM of Theorem 3.3.4(ii) (namely the unique matching with domain im ι such that χq ◦ ι = BN ).

In general, if f : M → N is a morphism of PFD persistence modules, then f factors canonically as qf

jf

→ → im(f ) − → N. M −− We define the partial matching χf : BM BN to be the composition χf = χj f ◦ χ qf .

Proposition 3.4.1.

Let f :M → N be a morphism of PFD persistence modules. Suppose χf b, d = b , d . Then b ≤ b < d ≤ d.

In words, induced matchings shift the endpoints of an interval to the left.

Proof.

Theorem 3.3.4, we have χqf b, d = b, d with d ≤ d and

By χjf b, d = b , d with b ≤ b. The middle inequality b < d holds because b, d is an interval.

Proposition 3.4.2. χ is functorial when restricted to the subcategory of monomorphisms of PFD persistence modules. Dually, χ is functorial when restricted to the subcategory of epimorphisms of PFD persistence modules.

Proof. Let j1 : M → N and j2 : N → P be monomorphisms of PFD persistence modules. By definition, BN is the disjoint χ j1 : B M union of the canonical injections ·, d M → ·, d N , and similarly for χj2 : BN BP and χj2 ◦j1 : BM BP . By Remark 3.3.1, it follows that χj2 ◦j1 = χj2 ◦ χj1 . Thus χ is functorial when restricted to the subcategory of monomorphisms in vectR .

The result for epimorphisms follows by essentially the same argument, together with the fact that the operation of reversing partial matchings is functorial. Thus χ is functorial when restricted to the subcategory of epimorphisms in vectR .

The Stability Theorem of Persistent Homology

39

For N a persistence module and > 0, we define a submodule N of N by setting Nt = im ϕN (t − , t)

for all t ∈ R, with transition maps the restriction of ϕN to N .

Definition 3.4.3. A persistence module N is said to be −trivial if the -transition morphism ϕ N : N → N ( ) is the zero morphism, or equivalently if N = 0.

Theorem 3.4.4 (Induced Matching Theorem). Let f : M → be a N morphism of PFD persistence modules, and suppose that χ b, d = f

b , d . Then (i) b ≤ b < d ≤ d.

⊂ im(χ ) and (ii) If coker(f ) = N/im(f ) is -trivial, then BN f

b ≤ b ≤ b + .

⊂ dom(χ ) and (iii) Dually, if ker(f ) is -trivial, then BM f

d − ≤ d ≤ d.

Proof. Part (i) is exactly Proposition 3.4.4.

Let N be as defined immediately before Definition 3.4.3, and recall is the collection of intervals in B containing a closed interval that BN N of length . First observe that to obtain BN , one simply shifts the left to the right by : endpoints of intervals in BN

}.

BN = { b + , d | b, d ∈ BN

Next, let j : N → the inclusion N be map. By the definition and of χj , we see that χj b + , d = b, d for every b, d ∈ BN

consequently that im(χj ) = BN .

To prove Part (ii), suppose that coker(f ) is -trivial. Then N ⊂ im(f ). (Indeed, this follows because ϕcoker(f ) (t − , t) = 0 if and only if im ϕN (t − , t) ⊂ im(ft ).) This implies that the following diagram commutes, where each map is the inclusion: N j

N

jf j

im(f )

40

Adam Gardner

By Proposition 3.4.2, Ď&#x2021;j = Ď&#x2021;jf â&#x2014;Ś Ď&#x2021;j . Furthermore, Ď&#x2021;f = Ď&#x2021;jf â&#x2014;Ś Ď&#x2021;qf by the definition of Ď&#x2021;f . Hence the following diagram commutes: BN

Ď&#x2021;j

Ď&#x2021;f

BN

Ď&#x2021;jf

BM

Ď&#x2021;j Ď&#x2021;qf

Bim(f ) = im(Ď&#x2021; ) â&#x160;&#x201A; im(Ď&#x2021; ). By By the commutativity of the left triangle, BN j jf the definition of the induced partial matchings, im(Ď&#x2021;jf ) = im(Ď&#x2021;f ) and so

â&#x160;&#x201A; im(Ď&#x2021;f ), BN

as claimed.

b, d = To finish the proof of Part (ii), we must show that whenever Ď&#x2021; f

b , d the inequality b â&#x2030;¤ b â&#x2030;¤ b + holds. The lefthand inequality follows from Part (i), and the righthand inequality follows by the commutativity of the left triangle as follows.

b, d = By the definition of the induced partial matching, we have Ď&#x2021; j f

b , d . Now Ď&#x2021;j b + , d = b , d , so by the commutativity of the

left triangle we have Ď&#x2021;j b + , d = b, d . The inequality b â&#x2030;¤ b + therefore follows by applying Theorem 3.3.4 to the monomorphism j : N â&#x2020;&#x2019; N .

Finally, the proof of Part (ii) dualizes readily to a proof of Part (iii).

3.5

An Explicit Formulation of the Algebraic Stability Theorem

We shall see that the Algebraic Stability Theorem (Theorem 2.3.15 above and the stronger Theorem 3.5.2 below) follows readily from the Induced Matching Theorem (Theorem 3.4.4).

Lemma 3.5.1. If f : M â&#x2020;&#x2019; N (δ) is a δ-interleaving morphism, then both ker(f ) and coker(f ) are 2δ-trivial.

The Stability Theorem of Persistent Homology

41

Proof. By the definition of a δ-interleaving, there exists a morphism g : N → M (δ) such that g(δ) ◦ f = ϕ2δ M and f (δ) ◦ g = ϕ2δ N. The first equality implies that ker(f ) is 2δ-trivial, and the second equality implies that coker(f ) is 2δ-trivial.

For a persistence module N and δ ≥ 0, there is a bijection r δ : BN (δ) → BN given by rδ b, d = b + δ, d + δ for each interval b, d ∈ BN (δ) .

Theorem 3.5.2 (Explicit Formulation of the Algebraic Stability Theorem). If f : M → N (δ) is a δ-interleaving of PFD persistence modules, then rδ ◦ χf : BM BN is a δ-matching. In particular, dB (BM , BN ) ≤ dI (M, N ).

Proof. By Lemma 3.5.1, ker(f ) and coker(f ) are both 2δ-trivial. By the 2δ 2δ Induced Matching Theorem,

we have that BM ⊂ dom(χf ) and BN ⊂ im(χf ). If χf b, d = b , d , we have that b ≤ b ≤ b + 2δ and d − 2δ ≤ d ≤ d,

or equivalently, in terms of the endpoints of rδ ◦ χf b, d = b + δ, d + δ , (b + δ) − δ ≤ b ≤ (b + δ) + δ and d − δ ≤ (d + δ) ≤ d + δ.

This verifies that rδ ◦ χf is indeed a δ-matching.

Combining Theorem 2.3.14 (The Converse Stability Theorem for Decomposable Persistence Modules) and 2.1.7 (every PFD persistence module is decomposable), recalling that dB (BM , BN ) = dB (dgm(M ), dgm(N )) for persistence modules M and N , we see that the inequality of Theorem 3.5.2 is actually an equality: Theorem 3.5.3 (The Isometry Theorem for PFD Persistence Modules). If M and N are PFD persistence modules, then dB (BM , BN ) = dI (M, N ).

42

Adam Gardner

This result concludes our study of persistence modules.

Acknowledgement I wish to thank my supervisors Maia Fraser and Yael Karshon for their exceptional guidance and feedback during the writing of this masterâ&#x20AC;&#x2122;s project. The topic and readings were suggested by Maia Fraser and I thank her for many technical discussions, including her suggestions to make the proof of Theorem 2.1.4 more direct. I am grateful to the Canadian Institutes of Health Research (CIHR), the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Social Sciences and Humanities Research Council of Canada (SSHRC) for the funding provided by their CGS-M scholarship. I appreciate the facilities and educational opportunities provided to me by University of Toronto during the completion of my masterâ&#x20AC;&#x2122;s degree.

Adam Gardner Department of Mathematics, University of Toronto, Address, adam.gardner@mail.utoronto.ca

References [1] G. Azumaya. Corrections and supplementaries to my paper concerning KrullRemakSchmidts theorem. Nagoya Mathematical Journal, 1:117124, 1950.

[2] U. Bauer and M. Lesnick. Induced matchings and the algebraic stability of persistence barcodes. Preprint, 2015. arXiv:1311.3681.

[3] G. Carlsson, A. Collins, L. Guibas and A. Zomorodian. Persistence barcodes for shapes. Proceedings of the 2nd Symposium on Geometry Processing, 127138, 2004.

[4] F. Chazal, D. C. Steiner, M. Glisse, L. Guibas, and S. Oudot. Proximity of persistence modules and their diagrams. Proceedings of the 25th Annual Symposium on Computational Geometry, 237246, ACM, 2009.

[5] F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The structure and stability of persistence modules. Preprint, 2012. arXiv:1207.3674.

[6] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103120, 2007.

[7] W. Crawley-Boevey. Decomposition of pointwise finite-dimensional persistence modules. Preprint, 2012. arXiv:1210.0819.

[8] H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete & Computational Geometry, 28(4):511533, 2002.

The Stability Theorem of Persistent Homology

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[9] M. Lesnick. The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics, 2015. To appear. arXiv:1106. 5305.

Morfismos, Vol. 21, No. 1, 2017, pp. 45–67 Morfismos, Vol. 21, No. 1, 2017, pp. 45–67

Aproximaci´on m´etrica de grupos: una breve ∗ Aproximaci´onperspectiva m´etrica de grupos: una breve Luis Manuel Rivera Luis Manuel Rivera

∗ perspectiva Nidya Monserrath Veyna Garc´ıa Nidya Monserrath Veyna Garc´ıa Resumen

Los grupos s´ oficos y los grupos hiperlineales han generado una gran cantidad de investigaci´ on Resumen en los u ´ltimos a˜ nos en diversas areasLos ´ de grupos matem´ as´ ticas tales como teor´ ıa geom´ e trica grupos, una oficos y los grupos hiperlineales handegenerado din´ amica simb´ o lica y a ´ lgebra de operadores.

Adem´ a s, los grupos gran cantidad de investigaci´ on en los u ´ltimos a˜ nos en diversas s´ oficos han ganado inter´ e s porque se ha demostrado que cumplen areas de matem´ ´ aticas tales como teor´ıa geom´etrica de grupos, varias conjeturas a´ n abiertas para grupos enAdem´ general. din´ amica simb´ oulica y´ algebra de los operadores. as, losLas grupos definiciones de ambas gruposseson alogas y seque pueden s´ oficos han ganadoclases inter´ede s porque haan´ demostrado cumplen pensar comoconjeturas dentro de a´ una de grupos de grupos recienteen estudio que Las varias un clase abiertas para los general. se conocen como de losambas gruposclases que tienen la propiedad de aproximadefiniciones de grupos son an´alogas y se pueden ci´ on pensar m´etrica.

Endentro este art´ se presenta un de panorama como deıculo una clase de grupos reciente general estudio que de dicha clase de grupos. se conocen como los grupos que tienen la propiedad de aproximaci´ on m´etrica. En este art´ıculo se presenta un panorama general

2010 Mathematics Subject Classification: 20-02, 20F65, 20E22, 43A07. de dicha clase de grupos.

Keywords and phrases: Ultraproductos, grupos s´ oficos, grupos hiper2010aproximaci´ Mathematics Classification: 20-02, 20F65, 20E22, 43A07. lineales, on Subject de grupos.

Keywords and phrases: Ultraproductos, grupos s´ oficos, grupos hiperlineales, aproximaci´ on de grupos.

1

Introducci´ on

El concepto de ultraproducto 1 Introducci´ on aparece de manera general en un art´ıculo de Jerzy L oˇs en 1955 [20], pero fue en a˜ nos recientes cuando ha tomado El concepto de ultraproducto maneraLo general en porque un art´ıculo importancia en ciertas areas de la aparece ´ teor´ıa dedegrupos. anterior de demostrado Jerzy L oˇs enque 1955varias [20], pero en a˜ nos recientes cuando ha tomado se ha clasesfueimportantes de grupos tienen la importancia enunciertas areas de la teor´ ıa de grupos.

Lo anterior porque caracter´ ıstica com´ de ser´ isomorfos a subgrupos de ciertos ultraprose ha clases importantes ductos de demostrado grupos, y enque estevarias caso decimos que el grupodeengrupos cuesti´otienen n se la ıstica com´ un de ser isomorfos a subgrupos de ciertos ultrapro∗ caracter´ Este art´ıculo es parte de la tesis de licenciatura de la segunda autora bajo ductos de grupos, y en La estetesis caso que grupoAcad´ en ecuesti´ on se la direcci´ on del primer autor. se decimos present´ o en la el Unidad mica de Matem´ a∗ticas de la Universidad Aut´ onoma de Zacatecas en marzo del 2016 como Este art´ıculo es parte de la tesis de licenciatura de la segunda autora bajo requisito para obtener el t´ıtulo de Licenciada en Matem´ aticas. la direcci´ on del primer autor. La tesis se present´ o en la Unidad Acad´emica de Matem´ aticas de la Universidad Aut´ onoma de Zacatecas en marzo del 2016 como requisito para obtener el t´ıtulo de Licenciada en Matem´ aticas.

45

45

46

Luis Rivera y Nidya Veyna

aproxima por los grupos con los cuales se construye el ultraproducto.

Fue en los a˜ nos noventa cuando se comenzaron a definir estas clases de grupos, siendo los grupos s´ oficos y los grupos hiperlineales dos de las clases m´ as importantes, las cuales surgen en diferentes ramas de la matem´atica (din´ amica simb´ olica y ´ algebra de operadores, respectivamente). Estos grupos han ganado inter´es porque se ha demostrado que cumplen varias conjeturas importantes que siguen abiertas para grupos en general.

La clase de grupos s´ oficos fue definida, sin usar ultraproductos, por Gromov [15], quien demostr´ o que esta clase de grupos satisfacen la conjetura de sobreyuntividad de Gottschalk [16]. El nombre de grupos s´oficos se debe a Weiss [37]. Estos grupos son una generalizaci´on com´ un de los grupos residualmente finitos y de los grupos amenables, y tambi´en incluyen, entre otros, a los grupos encajables localmente en finito, estos u ´ltimos presentados por Vershik y Gordon [14].

La clase de grupos hiperlineales fue definida por R˘adulescu [28] quien demostr´o que estos grupos cumplen la conjetura del encaje de Connes en su versi´on para grupos. Elek y Szab´o [9] demostraron que los grupos s´oficos son hiperlineales al mostrar que se pueden encajar en un cierto ultraproducto m´etrico de grupos sim´etricos finitos, lo que implica que los grupos s´ oficos cumplen la conjetura del encaje de Connes.

Elek y Szab´ o [8] tambi´en demuestran que los grupos s´oficos cumplen con la conjetura de finitud directa de Kaplansky y con la conjetura del determinante de L¨ uck [9]. Posteriormente, Thom [35] demostr´o que dichos grupos tamb´ıen cumplen con la conjetura de autovalores algebraicos de J. Dodziuk, P. Linnell, V. Mathai, T. Schick y S. Yates. A la fecha no se conocen ejemplos de grupos que no sean s´oficos o hiperlineales. M´as detalles sobre estas dos clases de grupos se pueden consultar, por ejemplo, en el resumen de Pestov [27], en el libro de Capraro y Lupini [3] y en la monograf´ıa de Ceccherini-Silberstein y Coornaert [6].

A partir de entonces, se han definido otras clases de grupos como posibles generalizaciones de los grupos s´oficos: los grupos s´oficos d´ebiles, los grupos s´ oficos lineales y los grupos K-lineales. El siguiente diagrama muestra la relaci´ on entre algunas de estas clases de grupos: Finito ⇓ Amenable

⇒ ⇒

Res. Finito ⇓ Res. Amenable

⇒ ⇒

LEF ⇓ LEA

=⇒

S´ ofico Lineal ⇑ S´ ofico

⇒

S´ ofico D´ebil

⇒

Hiperlineal

Aproximaci´ on m´etrica de grupos

47

Los grupos s´ oficos, los grupos s´ oficos d´ebiles, los grupos s´oficos lineales y los grupos K-lineales se pueden definir sin usar ultraproductos, mediante definiciones que tiene muchas similitudes. La diferencia principal entre las respectivas definiciones para cada tipo de grupos, radica en la clase de grupos m´etricos que aproximan al grupo en cuesti´on.

Por ejemplo, los grupos s´ oficos se aproximan por grupos sim´etricos finitos equipados con la m´etrica de Hamming y los grupos hiperlineales se aproximan por la clase de grupos unitarios de rango finito equipados con la m´etrica de Hilbert-Schmidt.

En este art´ıculo se presenta un breve panorama de esta clase de grupos. Nuestro tratamiento del tema es limitado e incompleto pero tiene como objetivo presentar el ´ area de investigaci´on sobre la aproximaci´ on de grupos de manera breve.

A pesar de que la literatura sobre este tema est´a en constante crecimiento y se cuenta con al menos tres fuentes en donde se resumen algunos de los resultados del ´ area ([6, 3, 27]), en la actualidad, a conocimiento de los autores, no se cuenta con literatura en espa˜ nol sobre este tema. Adem´ as, a diferencia de [6, 3, 27], en este art´ıculo se aborda el tema de aproximaci´ on de manera general como se ha hecho recientemente por varios autores [12, 18, 19, 33, 36, 34]. La mayor´ıa de resultados se presentan sin demostraci´ on pero se indica las fuentes en donde se pueden consultar.

El resto del art´ıculo est´ a organizado como sigue. En la secci´on 2 se presenta un breve resumen de definiciones y resultados sobre ultrafiltros. En la secci´ on 3 se presentan a los ultraproductos y se define a los grupos que son localmente encajables en una clase de grupos por medio de dos definiciones, una de las cuales usa ultraproductos. En la secci´on 4 se define a los ultraproductos m´etricos y se definen el concepto de aproximaci´ on m´etrica de grupos. Posteriormente se dan varios ejemplos de esta clase de grupos: los grupos s´oficos, los grupos hiperlineales, los grupos s´ oficos d´ebiles y los grupos K-lineales. En la secci´on 5 se extiende la exposici´ on sobre los grupos s´oficos, que es la clase m´as estudiada en la teor´ıa de aproximaci´ on de grupos. Se presentan varias definiciones equivalentes de grupos s´ oficos, algunos ejemplos y algunas propiedades de cerradura de esta clase de grupos. En la secci´on 7 se presenta una segunda definici´ on de grupos aproximables que no usa ultrafiltros, esta definici´ on es an´ aloga a la de grupos s´oficos. Adem´as se presentan algunos resultados de cerradura de esta clase de grupos.

48

2

Luis Rivera y Nidya Veyna

Ultrafiltros

Vamos a presentar una breve introducci´on a la teor´ıa de ultrafiltros.

Esta secci´on esta basada principalmente en [1, cap´ıtulo 2], [6, ap´endice J] y [38, cap´ıtulo 1].

Definici´ on 2.1. Sea X un conjunto no vac´ıo. Un filtro F en X es una familia no vac´ıa de subconjuntos de X que satisface F1) ∅ ∈ / F; F2) Si A, B ∈ F entonces A ∩ B ∈ F; F3) Si A ∈ F y A ⊆ B entonces B ∈ F. Algunos autores, por ejemplo en [1], no incluyen la condici´on (F1) en la definici´ on de filtro. Notemos que si usamos u ´nicamente las condiciones (F2) y (F3) en la definici´ on anterior, sigue que ∅ ∈ F si y solo si F = P(X). En este caso, al conjunto P(X) se le llama filtro impropio.

Los siguientes son ejemplos de filtros: Ejemplo 2.2. (a) Para x0 ∈ X, la familia Fx0 = {A ⊆ X : x0 ∈ A} es un filtro en X y se llama filtro principal generado por x0 . (b) Sea X un conjunto infinito. La familia F = {A ⊆ X : X \ A es finito}, es un filtro en X y se llama el filtro de Fr´echet en X. Notemos que este filtro no es principal. (c) Sea T una topolog´ıa en X, y x ∈ X. El conjunto N (x) = {V : V es una vecindad de x} es un filtro en X. A este filtro se le conoce como el filtro de vecindades de x. (d) Sea (X, ≤) un conjunto dirigido no vac´ıo. Un subconjunto A ⊆ X se llama residual en X si existe x0 ∈ X tal que el conjunto {x ∈ X : x0 ≤ x} es un subconjunto de A. El conjunto Fr (X) de todos los subconjuntos residuales de X es un filtro en X y se llama el filtro residual en X.

Aproximaci´ on m´etrica de grupos

49

Notemos que si consideramos al conjunto dirigido (N, ≤), un subconjunto A ⊆ N es residual si y solo si N \ A es finito. Entonces, se tiene que el filtro residual en N es el filtro de Fr´echet en N. Proposici´ on 2.3. Un ultrafiltro U en X es principal si y solo si existe un conjunto finito en U .

Definici´ on 2.4. Sea X un conjunto. Decimos que una familia A de subconjuntos de X tiene la propiedad de intersecci´ on finita, o que es un sistema centrado, si la intersecci´ on de cualquier subcolecci´on finita de A es no vac´ıa.

Proposici´ on 2.5. Sea X un conjunto no vac´ıo y sea A ⊂ P(X). Entonces, existe un filtro en X que contiene a A si y solo si A tiene la propiedad de intersecci´ on finita.

Definici´ on 2.6. Un filtro F en X se llama ultrafiltro si es un filtro maximal (relativo a la inclusi´ on).

Proposici´ on 2.7. Un filtro U en X es un ultrafiltro si y solo si para todo A ⊆ X, o bien A ∈ U , o bien X \ A ∈ U .

Definici´ on 2.8. Un ultrafiltro U es libre, si sus elementos no tienen puntos en com´ un, es decir, si U = ∅.

Observemos que cualquier filtro principal es un ultrafiltro. Si consideramos a X = N tenemos que el filtro de Fr´echet en N no es un ultrafiltro porque ninguno de los conjuntos 2N o N \ 2N pertenecen a dicho filtro, sin embargo se sabe que cualquier ultrafiltro libre sobre X contiene al filtro de Fr´echet. La siguiente proposici´on requiere del Lema de Zorn.

Teorema 2.9. Sea X un conjunto no vac´ıo. Cualquier filtro en X es un subconjunto de alg´ un ultrafiltro en X. Proposici´ on 2.10. Todo ultrafiltro es, o bien principal, o bien libre.

La existencia de los ultrafiltros libres es una consecuencia del Lema de Zorn. Por ejemplo, considere la siguiente colecci´on de subconjuntos de N+ C = {N+ \ {n} : n ∈ N+ }.

Esta colecci´ on de conjuntos tiene la propiedad de intersecci´on finita, y es tal que C = ∅. As´ı por la proposici´on 2.5 y el teorema 2.9, existe un ultrafiltro U que lo contiene. Adem´as, ∩U ⊆ ∩C = ∅. Por lo tanto U es un ultrafiltro libre.

50

Luis Rivera y Nidya Veyna

Definici´ on 2.11. Sea X un conjunto, Y un espacio topol´ogico, y0 ∈ Y , f : X → Y una funci´on y F un filtro en X. Se dice que y0 es un l´ımite de f a lo largo de F (o que f (x) converge a y0 a lo largo de F), si f −1 (V ) = {x ∈ X : f (x) ∈ V } pertenece a F para todo vecindad V de as F es un ultrafiltro decimos que y0 es un ultral´ımite de f y0 . Si adem´ a lo largo de F. Si y0 es un l´ımite u ´nico se denota por lim f (x) = y0 , o simplemente por lim f (x) = y0 .

F

x→F

Proposici´ on 2.12. Sean X un conjunto, Y un espacio topol´ ogico compacto, f : X → Y una funci´ on, y U un ultrafiltro en X. Entonces existe as, si Y es y0 ∈ Y tal que f (x) converge a y0 a lo largo de F. Adem´ ´nico.

Hausdorff, entonces y0 es u La definici´ on de ultral´ımite para el caso de series de n´ umeros reales, queda como sigue: umeros reales y U un Definici´ on 2.13. Sea {xi }i∈I una familia de n´ ultrafiltro en I. Decimos que lim xi = x ∈ R si para todo > 0 se tiene u

{i ∈ I : |xi − x| < } ∈ U El c´alculo del ultral´ımite depende de la selecci´on del ultrafiltro. Por ejemplo, sea {xn }n∈N una secuencia convergente de n´ umeros reales. Sea U un ultrafiltro principal en N generado por {n0 }, se puede demostrar que lim xn = xn0 . u

Ahora, sea U un ultrafiltro libre en N, sea x el l´ımite cl´asico de {xn }n∈N , se puede demostrar que lim xn = x. u

3

Ultraproductos de grupos

Vamos a presentar una estructura algebraica que se construye como cociente de un producto directo P de una familia de grupos (Gi )i∈I . Donde el cociente est´ a dado por una cierta relaci´on en P que se define usando un filtro F en I. Cuando el filtro F es ultrafiltro, a dicha estructura se le conoce como ultraproducto. Adem´as, vamos a presentar una clase de grupos, en donde cada grupo de dicha clase tiene la propiedad de poderse encajar en un cierto ultraproducto. Esta secci´on est´a basado principalmente en la monograf´ıa de Ceccherini-Silberstein y Coornaert [6, secci´on 7].

Aproximaci´ on m´etrica de grupos

51

Sea (Gi )i∈I una familia de grupos y F un filtro en el conjunto de ´ındices I. Sea P el producto directo de la familia (Gi )i∈I , es decir

P = Gi . i∈I

Sean g = (gi )i∈I y h = (hi )i∈I elementos de P . Decimos que g ∼F h si el conjunto {i ∈ I : gi = hi } pertenece a F. Esta relaci´on es de equivalencia como se muestra en los puntos (1)-(3) de la siguiente proposici´on.

Proposici´ on 3.1. Para todo a, b, c, d ∈ P se tiene que: 1. a ∼F a;

2. a ∼F b si y solo si b ∼F a; 3. Si a ∼F b y b ∼F c entonces a ∼F c; 4. Si a ∼F b y c ∼F d entonces ac ∼F bd; 5. a ∼F b si y solo si a−1 ∼F b−1 .

Con los puntos (3) y (4) de la proposici´on anterior se puede demostrar la siguiente Proposici´ on 3.2. Sea NF = {g ∈ P : g ∼F 1}, en donde 1 = (1Gi )i∈I , con 1G la identidad del grupo G. Entonces NF es un subgrupo normal de P .

Como NF es un grupo normal de P , el grupo cociente PF = P/NF est´a bien definido. Este grupo se llama producto reducido de la familia de grupos (Gi )i∈I con respecto al filtro F. Si F es un ultrafiltro, decimos que PF es el ultraproducto de la familia de grupos (Gi )i∈I con respecto al ultrafiltro F. La siguiente proposici´ on implica que el conjunto cociente P/∼F y el conjunto P/NF son iguales.

Proposici´ on 3.3. Si g, h ∈ P , entonces gNF = hNF ⇐⇒ g ∼F h.

Cuando F es un filtro principal, el grupo P/NF no es interesante en el sentido de que es isomorfo a un grupo en la familia (Gi )i∈I como se muestra a continuaci´ on. Sea Fk el filtro principal en I generado por k, entonces g ∼Fk 1 si y solo si gk = 1Gk . Por lo tanto NFk = {g ∈ P : gk = on φ : P → Gk como φ(g) = gk , tenemos que 1Gk }. Si definimos la funci´ φ es un homomorfismo sobreyectivo con ker(φ) = NFk . Por el primer teorema de homomorfismo de grupos obtenemos que P/NFk Gk .

52

Luis Rivera y Nidya Veyna

3.1

Grupos encajables localmente

En esta secci´ on vamos a definir a los grupos encajables localmente como primer ejemplo de grupos que se pueden aproximar, en un cierto sentido, por una clase de grupos. Primero algunas definiciones.

Definici´ on 3.4. Una colecci´ on de grupos G es una clase de grupos si satisface lo siguiente: si G ∈ G y G es un grupo isomorfo a G entonces G ∈ G .

Por ejemplo G puede ser la clase de grupos finitos, la clase de grupos nilpotentes, la clase de grupos solubles, etc.

Definici´ on 3.5. Sean G y H dos grupos. Dado un subconjunto finito F ⊆ G, una funci´ on φ : G → H se llama F -casi-homomorfismo de G en H si satisface: i) φ(f1 f2 ) = φ(f1 )φ(f2 ), para todo f1 , f2 ∈ F ; ii) φ|F es inyectiva.

Definici´ on 3.6. Sea G una clase de grupos. Se dice que un grupo Γ es localmente encajable en G si para todo subconjunto finito F ⊆ Γ, existe un grupo G ∈ G y un F -casi-homomorfismo ϕ de Γ en G. Por ejemplo, el grupo Z es localmente encajable en la clase de grupos finitos (grupos LEF por sus siglas en ingl´es) como se muestra a continuaci´on. Sea F un subconjunto finito de Z, elegimos a un entero n ≥ 0 tal que F ⊂ [−n, n]. Entonces, el homomorfismo cociente φ : Z → Z/(2n+1)Z es un F -casi-homomorfismo. M´as adelante veremos otros ejemplos de grupos LEF. Algunas propiedades de cerradura de esta clase de grupos son las siguientes.

Proposici´ on 3.7. Sea G una clase de grupos. Todo subgrupo de un grupo que es localmente encajable en G es localmente encajable en G .

Proposici´ on 3.8. Sea G una clase de grupos. Entonces Γ es localmente encajable en G si y solo si todo subgrupo finitamente generado de Γ es localmente encajable en G .

En el caso en que la clase G sea cerrada bajo productos directos tenemos que la encajabilidad local es cerrada al tomar producto directo.

Aproximaci´ on m´etrica de grupos

53

Proposici´ on 3.9. Sea G una clase de grupos la cual es cerrada bajo productos directos finitos. Sea (Γi )i∈I una familia de grupos que son localmente encajables en G . Entonces, su producto directo Γ = i∈I Γi es localmente encajable en G .

Los grupos localmente encajables son un ejemplo de grupos que se pueden definir usando ultraproductos como lo muestra el siguiente teorema.

Teorema 3.10. Sea G una clase de grupos y sea Γ un grupo. Las siguientes condiciones son equivalentes. a) Γ es localmente encajable en G ; b) Existe una familia de grupos (Gi )i∈I tal que Gi ∈ G , para todo i ∈ I, y un ultrafiltro U en I tal que Γ es isomorfo a un subgrupo del ultraproducto PU de la familia (Gi )i∈I con respecto al ultrafiltro U.

Si Γ es un grupo que cumple con la condici´on (b) del teorema anterior decimos que Γ es aproximado por los grupos en la familia (Gi )i∈I . Una consecuencia de este teorema es que la definici´on de grupos encajables localmente no depende de la selecci´ on particular del ultrafiltro.

Al parecer, Gordon y Vershik fueron los primeros en estudiar a los grupos encajables localmente en la clase de grupos finitos (LEF) [14].

En su trabajo ellos mostraron que las siguientes clases de grupos son LEF: grupos finitos, grupos libres, grupos abelianos, grupos nilpotentes, grupos de matrices. Ellos tambi´en demostraron que cualquier grupo finitamente presentado con problema de la palabra no decidible (Pyotr Novikov [24] demostr´ o que tales grupos existen) no es un grupo LEF. Adem´as, demostraron que el grupo G = b, t : t−1 b2 t = b3 es un grupo finitamente presentado con problema de la palabra decidible que no es LEF. En el libro de Magnus, Karrass y Solitar [21] se pueden consultar las definiciones y resultados b´asicos sobre el tema de presentaciones de grupos y sobre el problema de la palabra.

4

Aproximaci´ on m´ etrica de grupos

En esta secci´ on vamos a presentar otro tipo de aproximaci´on de grupos conocida como aproximaci´ on m´etrica. Esta secci´on est´a basada principalmente en los art´ıculos de Stolz [33, secci´on 2] y de Stolz y Thom

54

Luis Rivera y Nidya Veyna

[34, secci´on 2]. Ambos art´ıculos forman parte de la tesis doctoral de Stolz [32]. Vamos a utilizar funciones de longitud, en lugar de m´etricas, que es en la manera como lo han hecho recientemente diversos autores [12, 19, 32, 33, 36, 34].

Definici´ on 4.1. Sea G un grupo. Una funci´on : G → [0, 1] es una funci´on de longitud en G si para todo g, h ∈ G se cumple lo siguiente: FL1) (g) = 0 si y solo si g = 1; FL2) (g) = (g −1 ); FL3) (gh) ≤ (g) + (h).

Decimos que es invariante si adem´as se cumple (hgh−1 ) = (g), para todo h, g ∈ G. Notemos que esta condici´on es equivalente a pedir que (gh) = (hg) para todo h, g ∈ G. A una funci´on se le llama pseudo funci´ on de longitud si remplazamos la condici´on (FL1) por (1) = 0.

La funci´ on de longitud trivial 0 sobre un grupo G se define como 0 (g) = 1 si g = 1 y 0 (1) = 0. Las (pseudo) funciones de longitud est´an relacionadas con las (pseudo) m´etricas como sigue Proposici´ on 4.2. Sea una (pseudo) funci´ on de longitud en un grupo G. Entonces d(g, h) := (gh−1 ) define una (pseudo) m´etrica en G. Si es invariante entonces d es bi-invariante. Si por el contrario d es una (pseudo) m´etrica izquierdo-invariante en G entonces (g) := d(g, 1) es una (pseudo) funci´ on de longitud. Si d es bi-invariante entonces es invariante.

Algunos ejemplos de funciones de longitud son: 1. Sea [n] el conjunto de los naturales {1, . . . , n}. Denotemos por Sn al grupo sim´etrico Sym([n]) que consiste de todas las funciones biyectivas de [n] sobre [n] con la operaci´on de grupo dada por la composici´ on de funciones. La funci´on de longitud de Hamming de una permutaci´ on σ ∈ Sn , se define como H (σ) =

|{i ∈ [n] : σ(i) = i}| . n

2. El grupo Un de elementos unitarios de Mn (C) se puede equipar con la funci´ on de longitud invariante de Hilbert-Schmidt definida como HS (g) = dHS (g, 1),

Aproximaci´ on m´etrica de grupos

55

τ ((A − B)∗ (A − B)), en donde τ es la traza n

cij , y C ∗ es la matriz normalizada en Un (C) dada por τ (C) = n1

con dHS (A, B) =

i=1

transpuesta conjugada de C, para C ∈ Un (C).

3. En GL(n, C), el grupo de las matrices de n × n invertibles sobre C, se define la funci´ on de longitud del rango L (A) =

1 rango(I − A). n

4. Para cualquier grupo finito G se puede definir la pseudo funci´on de longitud conjugaci´ on dada por c (g) =

log|C(g)| log|G|

donde C(g) denota la clase de conjugaci´on de g.

La siguiente proposici´ on muestra una manera de definir una pseudo funci´on de longitud invariante en el cociente de un grupo que tiene pseudo funci´ on de longitud invariante.

Proposici´ on 4.3. Sea G un grupo con pseudo funci´ on de longitud invariante y H un subgrupo normal de G. Entonces G/H (gH) := inf (gx) x∈H

define una pseudo funci´ on de longitud invariante en G/H. Si G es finito y es una funci´ on de longitud, entonces G/H es una funci´ on de longitud.

4.1

Ultraproductos m´ etricos de grupos

Ahora vamos a construir los ultraproductos m´etricos. Para ello necesitamos el siguiente resultado.

Proposici´ on 4.4. Sea G un grupo con pseudo funci´ on de longitud invariante . Entonces el conjunto N = {g ∈ G : (g) = 0} es un subgrupo normal de G. Adem´ as G/N (gN ) = (g), g ∈ G, y G/N define una funci´ on de longitud invariante en G/N .

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Luis Rivera y Nidya Veyna

Para que N sea un grupo normal es necesario que la pseudo funci´on de longitud sea invariante como lo muestra el siguiente ejemplo, que aparece en el resumen de Pestov [27]. Para este ejemplo vamos a utilizar m´etrica en lugar de funci´ on de longitud.

Ejemplo 4.5. Sea SN el grupo sim´etrico que consiste de todas las biyecciones de N sobre si mismo, equipado con la siguiente m´etrica: d(Ď&#x192;, Ď&#x201E; ) =

2â&#x2C6;&#x2019;i

iâ&#x2C6;&#x2C6;D

en donde D = {i â&#x2C6;&#x2C6; N : Ď&#x192;(i) = Ď&#x201E; (i)}, y usamos la convenci´on de que la suma es cero cuando D es el conjunto vac´Ĺo. Sea U un ultrafiltro libre sobre N. Sean Ď&#x192;, Ď&#x201E; â&#x2C6;&#x2C6; N SN , tales que Ď&#x192; = (Ď&#x192;i ), con Ď&#x192;i = (i, i + 1), y Ď&#x201E; = (Ď&#x201E;i ), con Ď&#x201E;i = (1, i) (las funciones Ď&#x192;i y Ď&#x201E;i est´an expresadas en notaci´on c´Ĺclica). Es decir, Ď&#x192; y Ď&#x201E; son dos sucesiones de transposiciones en SN . Puesto que d (Ď&#x192;i , 1) = 2â&#x2C6;&#x2019;i + 2â&#x2C6;&#x2019;(i+1) tenemos que lim d (Ď&#x192;i , 1) = u

0 y as´Ĺ Ď&#x192; â&#x2C6;&#x2C6; N . Pero, como Ď&#x201E;i Ď&#x192;i Ď&#x201E;iâ&#x2C6;&#x2019;1 = (1, i + 1) y d ((1, i + 1) , 1) = 2â&#x2C6;&#x2019;1 + 2â&#x2C6;&#x2019;(i+1) , entonces lim d Ď&#x192;iâ&#x2C6;&#x2019;1 Ď&#x201E;i Ď&#x192;i , 1 = 1/2, lo cual implica que u

Ď&#x201E;i Ď&#x192;i Ď&#x201E;iâ&#x2C6;&#x2019;1 â&#x2C6;&#x2C6; N . Por lo tanto N no es un subgrupo normal de N SN .

En la siguiente proposici´ on se define una pseudo funci´on de longitud invariante en el producto directo de grupos equipados con pseudo funci´on de longitud invariante.

Proposici´ on 4.6. Sean (Gi , i )iâ&#x2C6;&#x2C6;I una secuencia de grupos cada uno equipado con una pseudo funci´ on de longitud i invariante. Sea P =

G su producto directo.

Sea U un ultrafiltro libre sobre I. Para iâ&#x2C6;&#x2C6;I i cada g â&#x2C6;&#x2C6; P , definimos Ë&#x153; := lim i (gi ). (g) u

Entonces Ë&#x153; define una pseudo funci´ on de longitud invariante en P .

Por las proposiciones 4.4 y 4.6 obtenemos que el grupo cociente

Gi /N, (P )u := iâ&#x2C6;&#x2C6;I

Ë&#x153; est´a bien definido, con N = {g â&#x2C6;&#x2C6; P : (g) = 0}, y adem´as P/N (gN ) es Ë&#x153; una funci´ on de longitud invariante en P/N tal que P/N (gN ) = (g), para todo g â&#x2C6;&#x2C6; P .

Al grupo (P )u se lo conoce como el ultraproductro m´etrico de la odulo U. familia (Gi , i )iâ&#x2C6;&#x2C6;I m´

Aproximaci´ on m´etrica de grupos

4.2

57

Grupos G –aproximables

Vamos a dar una definici´ on de aproximaci´ on m´etrica de grupos que usa ultraproductos.

Definici´ on 4.7. Sea G una clase de grupos, en donde cada grupo en la clase G est´ a equipado con una pseudo funci´on de longitud invariante.

Decimos que un grupo Γ tiene la propiedad de G -aproximaci´on m´etrica si existe un conjunto de ´ındices I y un ultrafiltro U en I tal que Γ es isomorfo a un subgrupo de un ultraproducto m´etrico ( i∈I Gi )u , con grupos Gi ∈ G . En este caso decimos que el grupo Γ es aproximado de forma m´etrica por los grupos en la clase G .

Desde hace varios a˜ nos se han estudiado clases de grupos que tienen la propiedad de aproximaci´ on m´etrica. A estos grupos se les han dado diferentes nombres dependiendo de la clase de grupos G que los aproximan y de las m´etricas con las que cuentan los grupos en G .

Definici´ on 4.8. Si Γ tiene la propiedad de G -aproximaci´on, al grupo Γ se le llama: • Localmente encajable en finito si G es la clase de grupos finitos con la funci´ on de longitud trivial 0 (Gordon y Vershik, [14]).

• S´ofico si G es la clase de grupos sim´etricos finitos con la funci´ on de longitud de Hamming (Gromov, [15]).

• Hiperlineal si G es la clase de matrices unitarias de rango finito sobre C con la funci´ on de longitud de Hilbert-Schmidt (R˘adulescu, [28]).

• S´ofico d´ebil si G es la clase de todos los grupos finitos, en donde cada grupo est´ a equipado con alguna funci´on de longitud invariante (Glebsky y Rivera, [13]).

• S´ofico lineal si G es la clase de grupos lineales generales GL(n, C) con (A) = n1 rango(I − A) (Arzhantseva y P˘aunescu, [2]).

• K-s´ofico si G es la clase de grupos lineales generales GL(n, F ), para F un campo fijo, con la longitud rango (Stolz, [33]).

Los grupos s´ oficos est´ an relacionados con los grupos hiperlineales de la siguiente manera. A cada permutaci´on σ ∈ Sn le podemos asociar

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Luis Rivera y Nidya Veyna

una matriz Aσ de n × n de la siguiente manera: (Aσ )ij =

1 si σ(j) = i 0 en otro caso

La funci´ on σ → Aσ define un encaje de Sn al grupo unitario U (n). Se puede verificar que H (α) = 12 ( HS (Aα ))2 , para toda α en Sn . Usando lo anterior Elek y Szab´ o [9] demostraron que los grupos s´oficos son hiperlineales. Las otras clases se relacionan como sigue: los grupos LEF son s´oficos, los grupos s´ oficos son s´ oficos d´ebiles [13], los grupos s´oficos son s´oficos lineales, y los grupos s´ oficos lineales son s´oficos d´ebiles [2].

En todos los casos se desconoce si el rec´ıproco es verdadero. A la fecha no se conocen ejemplos de grupos que no sean s´oficos, o hiperlineales, o s´oficos d´ebiles, o s´ oficos lineales, o K-s´oficos, y el encontrar un ejemplo de un grupo que no cumpla con alguna de estas definiciones es uno de los problemas m´ as importantes en la teor´ıa de aproximaci´on de grupos.

En la siguiente secci´ on continuaremos con la exposici´on sobre grupos s´oficos.

5

Grupos s´ oficos

Los grupos s´ oficos, son una de las clases de grupos G -aproximables m´as estudiadas a la fecha y son una generalizaci´on com´ un de dos clases importantes de grupos: los grupos residualmente finitos y los grupos amenables. En esta secci´ on vamos a dar otra definici´on de grupos s´oficos que no usa ultraproductos y vamos a presentar algunas propiedades de cerradura de esta clase de grupos. La equivalencia de las dos definiciones la da el teorema 7.2.

Definici´ on 5.1. Sea G un grupo, K ⊆ G un subconjunto finito no vac´ıo, > 0 y F un conjunto finito no vac´ıo. Una funci´on ϕ : G → Sym(F ) se llama (K, )-casi-homomorfismo si satisface las siguientes condiciones: i) para todo g, h ∈ K, dH (ϕ(g)ϕ(h), ϕ(gh)) ≤ ; ii) para todo g, h ∈ K, g = h, se tiene dH (ϕ(g), ϕ(h)) ≥ 1 − .

Definici´ on 5.2. Un grupo Γ es llamado s´ofico si para todo K ⊆ Γ subconjunto finito y para todo > 0, existe un conjunto finito no vac´ıo F y un (K, )-casi-homomorfismo ϕ : G → Sym(F ).

Aproximaci´ on m´etrica de grupos

59

Para el caso de grupos finitamente generados esta definici´on de grupos s´oficos es equivalente a la definici´ on original presentada por Gromov [15] y Weiss [37] (una demostraci´ on se puede consultar, por ejemplo, en [6, secci´on 7.7]). La siguiente proposici´on aparece en [25, cap´ıtulo 6] y es una recopilaci´ on de otras definiciones equivalentes de grupos s´oficos que han aparecido a lo largo de la literatura.

Proposici´ on 5.3. Sea Γ un grupo. Entonces las siguientes condiciones son equivalentes: 1. Para cada subconjunto finito K ⊆ Γ y para todo > 0, existe un conjunto finito no vac´ıo F y una funci´ on φ : Γ → Sym(F ) con las propiedades: i) dH (φ(g)φ(h), φ(gh)) ≤ para cada g, h ∈ K;

ii) dH (φ(1Γ ), 1Sym(F ) ) ≤ ;

iii) dH (φ(g), φ(h)) ≥ 1 − para cada g, h ∈ K con g = h.

(El grupo Γ tiene la propiedad de G -aproximaci´ on discreta fuerte) 2. Para cada subconjunto finito K ⊆ Γ y para todo > 0, existe un conjunto finito no vac´ıo F y una funci´ on φ : Γ → Sym(F ) con las propiedades: i) dH (φ(g)φ(h), φ(gh)) ≤ para cada g, h ∈ K;

ii) dH (φ(g), φ(h)) ≥ 1 − para cada g, h ∈ K con g = h.

3. Para cada constante δ ∈ (0, 1), cada subconjunto finito K ⊆ Γ y todo > 0, existe un conjunto finito no vac´ıo F y una funci´ on φ : Γ → Sym(F ) con las propiedades: i) dH (φ(g)φ(h), φ(gh)) ≤ para cada g, h ∈ K;

ii) dH (φ(g), φ(h)) ≥ δ para cada g, h ∈ K con g = h.

(El grupo Γ tiene la propiedad de G -aproximaci´ on discreta) 4. Existe alg´ un δ > 0 tal que para cada subconjunto finito K ⊆ Γ y todo > 0, existe un conjunto finito no vac´ıo F y una funci´ on φ : Γ → Sym(F ) con las propiedades: i) dH (φ(g)φ(h), φ(gh)) ≤ para cada g, h ∈ K;

ii) dH (φ(g), φ(h)) ≥ δ para cada g, h ∈ K con g = h. (Γ tiene la propiedad de G -aproximaci´ on)

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5. Para cada conjunto finito K ⊆ Γ, existe un δK tal que para cada > 0, existe un subconjunto finito no vac´ıo F y una funci´ on ϕ : Γ → Sym(F ) con las siguientes propiedades: i) dH (φ(g)φ(h), φ(gh)) ≤ para cada g, h ∈ K;

ii) dH (φ(g), φ(h)) ≥ δK para cada g, h ∈ K con g = h.

Definici´ on 5.4. Un grupo Γ se llama s´ ofico si cumple alguna (y por lo tanto todas) de las condiciones (1)-(5) de la proposici´ on anterior.

Teorema 5.5. La clase de grupos s´ oficos es cerrada con respecto a las siguientes operaciones: 1. Subgrupos [10]; 2. Producto directo [10]; 3. L´ımites directos [10]; 4. L´ımites inversos [10]; 5. Productos libres [10]; 6. Producto gr´ afico [4]; 7. Producto trenzado [17]; 8. Producto libre amalgamado sobre grupos amenables [11, 26] ; 9. Extensiones por grupos amenables: si N G, N es s´ ofico y G/N es amenable, entonces G es s´ ofico [10]; 10. Extensiones HNN sobre grupos amenables [5, 11, 26].

6

Ejemplos de grupos s´ oficos

Los grupos s´ oficos generalizan a dos clases importantes de grupos: los grupos residualmente finitos y los grupos amenables.

Definici´ on 6.1. Un grupo Γ es residualmente finito si para cada elemento g ∈ Γ con g = 1Γ , existe un grupo finito G y un homomorfismo φ : Γ → G tal que φ(g) = 1G .

Aproximaci´ on m´etrica de grupos

61

La clase de grupos residualmente finitos contienen a todos los grupos finitos, a los grupos abelianos finitamente generados (en particular el grupo aditivo Z), y a los grupos libres, entre otros. La clase de grupos residualmente finitos es cerrada al tomar subgrupos y l´ımites inversos.

Un teorema de Mal’cev [22] muestra que todos los grupos lineales finitamente generados son residualmente finitos. Los grupos aditivos Q, R y C no son grupos residualmente finitos. En el libro [6, secci´on 2] se puede consultar las demostraciones de lo dicho anteriormente, y otras propiedades de esta clase de grupos. Dos problemas importantes que permanecen abiertos sobre este tema son los siguientes: 1) determinar si todos los grupos Gromov-hiperb´ olicos son residualmente finitos; 2) determinar bajo que condiciones un grupo con un solo relator es residualmente finito [29].

Proposici´ on 6.2. Todo grupo residualmente finito es s´ ofico.

Definici´ on 6.3. Un grupo discreto Γ es amenable si para todo subconjunto finito Λ ⊂ Γ y para todo > 0 existe un conjunto E que es (Λ, )-Flner, es decir, un subconjunto finito E de Γ tal que para todo g∈Λ |gE E| < |E|, donde denota la diferencia sim´etrica entre conjuntos.

Cualquier grupo finito es amenable: sea Γ un grupo finito, entonces para cualquier subconjunto finito Λ ⊂ Γ y para todo > 0, el grupo Γ es un conjunto (Λ, )-Flner porque |gΓ∆Γ| = 0.

La clase de los grupos amenables contiene a los grupos finitos, a los grupos abelianos y a los grupos solubles, entre otros. El grupo Q es amenable pero no residualmente finito y el grupo libre con dos generadores F2 es residualmente finito pero no amenable (las demostraciones de estos hechos y m´ as informaci´ on de esta clase de grupos se puede consultar en [6, secci´ on 4]).

Proposici´ on 6.4. Todo grupo amenable es s´ ofico.

Finalizamos con el siguiente resultado que tiene como consecuencia que los grupos encajables localmente en la clase de grupos finitos (LEF) y los grupos encajables localmente en la clase de grupos amenables (LEA) son s´ oficos.

Proposici´ on 6.5. Todo grupo G que es localmente encajable en la clase de grupos s´ oficos es s´ ofico.

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Propiedades de grupos G -aproximables

En la secci´ on anterior se presenta una definici´on de grupos s´oficos que no utiliza ultrafiltros. Es una pregunta natural si es o no posible tener otra definici´ on de grupos G -aproximables que no use ultraproductos. En esta secci´on vamos a presentar dicha definici´on y veremos que es muy similar a la de grupos s´ oficos. Esta secci´on est´a basado principalmente en [19, 32, 33].

Definici´ on 7.1. Sea G una clase de grupos, cada uno de los cuales est´a equipado con una pseudo funci´ on de longitud . Un grupo Î&#x201C; tiene la propiedad de G -aproximaci´ on si para todo g â&#x2C6;&#x2C6; Î&#x201C;, g = 1, existe δg > 0 tal que para todo > 0 y cualquier subconjunto finito F â&#x160;&#x2020; Î&#x201C;, F = â&#x2C6;&#x2026;, existe un grupo G â&#x2C6;&#x2C6; G y una funci´ on Ď&#x2022; : Î&#x201C; â&#x2020;&#x2019; G tal que: GA1) (Ď&#x2022;(1)) â&#x2030;¤ ; GA2) (Ď&#x2022;(g)) â&#x2030;Ľ δg , para todo g â&#x2C6;&#x2C6; F \ {1}; GA3) (Ď&#x2022;(g)Ď&#x2022;(h)Ď&#x2022;(gh)â&#x2C6;&#x2019;1 ) â&#x2030;¤ , para todo g, h â&#x2C6;&#x2C6; F .

A la funci´ on Ď&#x2022; se le conoce como un (F, , δg , )-casi-homomorfismo.

Cuando no es necesario hacer referencia a δg y a la pseudo funci´on de longitud se dice simplemente que Ď&#x2022; es un (F, )-casi-homomorfismo.

Se asume que δg â&#x2030;¤ 1, para todo g â&#x2C6;&#x2C6; Î&#x201C; porque de lo contrario no se cumplir´Ĺa la condici´ on (GA2).

Un grupo que tiene la propiedad de G -aproximaci´on se llama s´ofico, hiperlineal, s´ ofico d´ebil, s´ ofico lineal, K-s´ofico dependiendo de quien es la clase G y la funci´ on de longitud de manera similar como en la definici´ on 4.8.

Una funci´ on δ : Î&#x201C; â&#x2020;&#x2019; R tal que δ(g) > 0, para g = 1 y δ(1) = 1 se le llama funci´ on de peso para el grupo Î&#x201C;. Notemos que en la definici´on de G -aproximaci´ on se requiere que el grupo Î&#x201C; tenga una funci´on de peso.

Otros tipos de aproximaci´ on de grupos son, la G -aproximaci´on discreta la cual se da al reemplazar en la definici´on δg por una constante δ; y la G -aproximaci´ on fuerte donde en la condici´on (GA2) se tiene (Ď&#x2022;(g)) â&#x2030;Ľ diam(G) â&#x2C6;&#x2019; para todo g â&#x2C6;&#x2C6; F \ {1}, en donde diam(G) = sup (g), y a Ď&#x2022; se le conoce como casi-homomorfismo fuerte. gâ&#x2C6;&#x2C6;G

No se sabe si las diferentes versiones de G -aproximaci´on son equivalentes de manera general. En el caso de grupos s´oficos e hiperlineales

Aproximaci´ on m´etrica de grupos

63

dichas versiones si son equivalentes. Para el caso de grupos s´oficos la equivalencia la da la proposici´ on 5.3 y para el caso de grupos hiperlineales la equivalencia la da un resultado que se puede encontrar, por ejemplo, en la tesis de Olesen [25, proposici´on 5.1.2].

El siguiente resultado, bien conocido en el ´area, muestra la equivalencia de las definiciones de aproximaci´ on 4.7 y 7.1 lo cual implica que la definici´ on de G -aproximaci´ on no depende de la selecci´on del ultrafiltro.

Teorema 7.2. Un grupo Γ tiene la propiedad de G -aproximaci´ on seg´ un la definici´ on 7.1 si y solo si existe un conjunto de ´ındices I y un ultrafil

tro U en I tal que Γ puede ser encajado en un ultraproducto i∈I Gi u de grupos Gi ∈ G .

7.1

Propiedades generales de grupos G -aproximables

Se conocen pocos resultados generales para los grupos G -aproximables.

Vamos a concluir este art´ıculo mencionando algunos de estos resultados.

Se sabe que la clase de grupos con la propiedad de G -aproximaci´on es cerrada al tomar subgrupos y l´ımites inversos [33, proposici´on 3.3]. Lo mismo es cierto para los grupos con la propiedad discreta y la propiedad discreta fuerte de G -aproximaci´ on. Se sabe que si la clase G tiene una propiedad conocida como amplificaci´ on, entonces la clase de grupos G aproximables es cerrada al tomar l´ımites directos [33, proposici´on 3.5].

Se conocen pocos resultados para el caso del producto directo de grupos G -aproximables. Stolz demostr´ o un resultado parcial con varias hip´otesis [33, proposici´ on 3.4]. Vamos a presentar el resultado que Derek F. Holt y Sarah Rees [19] anuncian en 2016.

Primero vamos a definir una funci´on de longitud en el producto directo de dos grupos con funci´ on de longitud. Si G es una clase de grupos m´etricos, escribiremos (G, G ) ∈ G para indicar que G ∈ G y on de longitud definida en G. que G es una funci´ Proposici´ on 7.3. Sea G una clase de grupos m´etricos. Supongamos que (G, G ), (H, H ) ∈ G . Entonces para p ∈ N ∪ {∞} y α ∈ G × H con a = (g, h), la funci´ on pG×H : G × H → [0, 1] definida por

p p p G (g) + H (h) p , p∈N G×H (g, h) = 2 y ∞ G×H (g, h) = max{ G (g), H (h)}

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es una funci´ on de longitud invariante.

Teorema 7.4. Sea G una clase de grupos con funciones de longitud invariantes asociadas y supongamos que, para alg´ un p â&#x2C6;&#x2C6; N â&#x2C6;Ş {â&#x2C6;&#x17E;} fijo, y para cualquier par de grupos G, H â&#x2C6;&#x2C6; G se cumple que (G, G ) , (H, H ) â&#x2C6;&#x2C6; G â&#x2021;&#x2019; (G Ă&#x2014; H, p ) â&#x2C6;&#x2C6; G .

Entonces el producto directo GĂ&#x2014;H de dos grupos G y H G -aproximables es G -aproximable.

El resultado de este teorema puede ser aplicado para deducir la cerradura bajo productos directos para las clases de grupos s´oficos d´ebiles, grupos LEF, grupos hiperlineales, grupos lineales s´oficos, donde la condici´on se mantiene de la siguiente manera: â&#x20AC;˘ Grupos s´ oficos d´ebiles para toda p; â&#x20AC;˘ Grupos LEF para p = â&#x2C6;&#x17E;; â&#x20AC;˘ Grupos hiperlineales para p = 2; â&#x20AC;˘ Grupos lineales s´ oficos para p = 1.

Para la funci´ on de longitud de Hamming G , H , la funci´on p G , h no es una funci´ on de longitud de Hamming, y por lo tanto no podemos deducir la cerradura de la clase de grupos s´oficos bajo el producto directo a partir de este resultado.

Concluimos este trabajo citando los art´Ĺculos de Holt y Ress [19], y de Hayes y Sale [18] en donde se pueden encontrar otras propiedades de cerradura para los grupos G -aproximables.

Agradecimientos Este art´Ĺculo es parte de la tesis de licenciatura de la segunda autora bajo la direcci´ on del primer autor. La tesis se present´o en la Unidad Acad´emica de Matem´ aticas de la Universidad Aut´onoma de Zacatecas en marzo del 2016. Los autores agradecen a los profesores Daniel Duarte, Patricia Jim´enez, Jes´ us LeaË&#x153; nos y Alexander Pyshchev por sus comentarios y sugerencias sobre la tesis, que se ven reflejados en este art´Ĺculo. Los autores agradecen al editor y al revisor por sus u ´tiles sugerencias y correcciones.

Luis Manuel Rivera Mart´Ĺnez Unidad Acad´emica de Matem´ aticas, Universidad Aut´ onoma de Zacatecas, Zacatecas, M´exico luismanuel.rivera@gmail.com

Nidya Montserrat Veyna Garc´Ĺa Unidad Acad´emica de Matem´ aticas, Universidad Aut´ onoma de Zacatecas, Zacatecas, M´exico nidyaveyna@gmail.com

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Morfismos se imprime en el taller de reproduccio ´n del Departamento de Matema ´ticas del Cinvestav, localizado en Avenida Instituto Polit´ecnico Nacional 2508, Colonia San Pedro Zacatenco, C.P. 07360, M´exico, D.F. Este nu ´mero se termino ´ de imprimir en el mes de junio de 2017. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

Contents - Contenido An approach to the topological complexity of the Klein bottle Donald M. Davis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The stability theorem of persistent homology Adam Gardner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Aproximacio ´n m´etrica de grupos: una breve perspectiva Luis Manuel Rivera y Nidya Monserrath Veyna Garc´ıa . . . . . . . . . . . . . . . . . . 45