COMPUTING WITH HOMOTOPY 1- AND 2-TYPES WHY? HOW? Graham Ellis NUI Galway, Ireland

Transcription

1 COMPUTING WITH HOMOTOPY 1- AND 2-TYPES WHY? HOW? Graham Ellis NUI Galway, Ireland

2 Why: 1. potential applications to data analysis 2. examples for generating theory How: 3. discrete Morse theory 4. homological perturbation

3 An n-type is a CW complex X with π i X = 0 for i > n.

4 An n-type is a CW complex X with π i X = 0 for i > n. A connected 1-type is modelled by the fundamental group π 1 X.

5 An n-type is a CW complex X with π i X = 0 for i > n. A connected 1-type is modelled by the fundamental group π 1 X. A connected 2-type is modelled by the homomorphism : π 2 (X,X 1 ) π 1 X 1 and action of π 1 X 1 on π 2 (X,X 1 ). The algebraic abstraction is a crossed module.

6 A filtered chain complex over a field C 1 C 2 C 3 C 4 induces for s t. ι s t n : H n (C s ) H n (C t ) The Persistent Betti numbers are β s t n = rank(ι s t n ) s t β s t n = 0 s > t.

7 β n bar code has β s,t n horizontal lines from column s to column t (β s t 2 ) =

8 1. Data Analysis Toy data points: v 1,v 2,...,v 72 R

9 1. Data Analysis Toy data points: v 1,v 2,...,v 72 R generated from

10 Fix a sequence of real numbers ǫ 1 < ǫ 2 < < ǫ T. The Rips simplicial complex X t has vertex set V = {v 1,...,v 72 }. n-simplices the subsets σ V with n+1 vertices and v v ǫ t for all v,v σ.

11 Persistent β 0 for C (X ):

12 Persistent β 1 for C (X ):

13 Data Model: A homotopy retract Y X 20

14 Data Model: A homotopy retract Y X 20 Y S 1 S 1

15 Different toy data: v 1,v 2,...,v 256 R 9

16 Different toy data: v 1,v 2,...,v 256 R 9 generated from

17 β 0 and β 1 bar codes for the 256 patches:

18 Data Model: A homotopy retract Y X 9 with π 1 Y = x,y xyx 1 y and just four (critical) cells e 0, e 1 1, e1 2, e2.

19 Data Model: A homotopy retract Y X 9 with π 1 Y = x,y xyx 1 y and just four (critical) cells e 0, e 1 1, e1 2, e2. Lyndon s theorem on one-relator groups implies π 2 Y = 0,

20 Data Model: A homotopy retract Y X 9 with π 1 Y = x,y xyx 1 y and just four (critical) cells e 0, e 1 1, e1 2, e2. Lyndon s theorem on one-relator groups implies and Y Klein Bottle. π 2 Y = 0,π n Y = 0 for n 2

21 Data Model: A homotopy retract Y X 9 with π 1 Y = x,y xyx 1 y and just four (critical) cells e 0, e 1 1, e1 2, e2. Lyndon s theorem on one-relator groups implies and Y Klein Bottle. π 2 Y = 0,π n Y = 0 for n 2 Caveat: [Kan-Thurston] For any space X there is a map K(G,1) X inducing H (K(G,1)) = H (X).

22 3-dimensional Euclidean data

23

24 Proposition: The alpha carbon atoms of the Thermus Thermophilus protein determine a knot K with peripheral system π 1 ( K) = a,b aba 1 b 1 π 1 (R 3 \K) = x,y xyx = yxy a x 2 yx 2 y b x. gap> K:=ReadPDBfile("1V2X.pdb"); Pure permutahedral complex of dimension 3 gap> Y:=RegularCWComplex(PureComplexComplement(K));; Regular CW-complex of dimension 3 gap> i:=boundary(y); Map of regular CW-complexes gap> phi:=fundamentalgroup(i,22495); [ f1, f2 ] -> [ f1^-3*f2*f1^2*f2*f1, f1 ]

25 A number of useful invariants are easily computed from a presentation of the fundamental group.

26 A number of useful invariants are easily computed from a presentation of the fundamental group. Computing low-index groups of a finitely presented group G Index n subgroup H G corresponds to a homomorphism G S n into the group of permutations of X = {gh g G}.

27 A number of useful invariants are easily computed from a presentation of the fundamental group. Computing low-index groups of a finitely presented group G Index n subgroup H G corresponds to a homomorphism G S n into the group of permutations of X = {gh g G}. Only finitely many such homomorphisms.

28 A number of useful invariants are easily computed from a presentation of the fundamental group. Computing low-index groups of a finitely presented group G Index n subgroup H G corresponds to a homomorphism G S n into the group of permutations of X = {gh g G}. Only finitely many such homomorphisms. Index n subgroups H G are finitely presented (Reidemeister- Schreier).

29 Proposition (Brendel, E, Juda, Mrozek ) The abelian invariants of subgroups G π 1 (R 3 \K) of index 7 distinguish between the ambient isotopy classes of prime knots with fourteen or fewer crossings, up to chirality.

30 A more quickly computed invariant: Construct the link K red K blue as a voxel image and then X = [0,1] 3 \(K red K blue ) as a CW complex with cells.

31 A more quickly computed invariant: Construct the link K red K blue as a voxel image and then X = [0,1] 3 \(K red K blue ) as a CW complex with cells. Compute π 1 X = x 1,x 2,x 3 : r 1,r 2,r 3,r 4.

32 A more quickly computed invariant: For generators α red,α blue H 1 (X,Z), β H 2 (X,Z) α red α blue = β r r 4π K red Kblue r r 3 dr dr = 6β

33 A more quickly computed invariant: For generators α red,α blue H 1 (X,Z), β H 2 (X,Z) α red α blue = β r r 4π K red Kblue r r 3 dr dr = 6β is instantly computed from P = x 1,x 2,x 3 : r 1,r 2,r 3,r 4 and : K(P) K(P) K(P).

34 For P = x,y : xyx = yxy the diagonal is given by: y x y y x y y x y y x y x x x x x y x x y x y y

35 For P = x,y : xyx = yxy the diagonal is given by: y x y y x y y x y y x y x x x x x y x x y x y y Only need orientations of quadrilaterals for cup product.

36 A more challenging invariant to compute: L = L = Y = R 3 \L Y = R 3 \L

37 A more challenging invariant to compute: L = L = Y = R 3 \L Y = R 3 \L H 3 ( π 1 Y/[[π 1 Y,π 1 Y],π 1 Y],Z) = Z 40 H 3 ( π 1 Y /[[π 1 Y,π 1 Y ],π 1 Y ],Z) = Z 6

38 2. A computation leading to some theory A crystallographic group is a discrete subgroup G < Isom(R n ) with finite index translation subgroup T G.

39 2. A computation leading to some theory A crystallographic group is a discrete subgroup G < Isom(R n ) with finite index translation subgroup T G. Set γ 1 T = T, γ i+1 T = [γ i T,G].

40 2. A computation leading to some theory A crystallographic group is a discrete subgroup G < Isom(R n ) with finite index translation subgroup T G. Set γ 1 T = T, γ i+1 T = [γ i T,G]. β s,t n = rank(h n (G/γ t T,F) H n (G/γ s T,F))

41 Illustration t: R R,x x +1 g: R R,x x G = t,g : g 2 = 1,tg = gt 1 = D

42 Illustration t: R R,x x +1 g: R R,x x G = t,g : g 2 = 1,tg = gt 1 = D G/γ n T = D 2 n 1

43 Illustration t: R R,x x +1 g: R R,x x G = t,g : g 2 = 1,tg = gt 1 = D G/γ n T = D 2 n 1 β 3 bar code

44 Proposition (E) Let G be an n-dimensional crystallographic group and let F = F p. We have H i (G/γ c T,F) = H i (G/γ c+d T,F) for all i 0,c k if:

45 Proposition (E) Let G be an n-dimensional crystallographic group and let F = F p. We have H i (G/γ c T,F) = H i (G/γ c+d T,F) for all i 0,c k if: P = G/T is a p-group;

46 Proposition (E) Let G be an n-dimensional crystallographic group and let F = F p. We have H i (G/γ c T,F) = H i (G/γ c+d T,F) for all i 0,c k if: P = G/T is a p-group; there exists an integer k 1 such that T/γ c+1 T is an n-generator abelian p-group for c k and, in the case p = 2, T/γ c+1 T is a direct product of cyclic groups of exponent > 2; and

47 Proposition (E) Let G be an n-dimensional crystallographic group and let F = F p. We have H i (G/γ c T,F) = H i (G/γ c+d T,F) for all i 0,c k if: P = G/T is a p-group; there exists an integer k 1 such that T/γ c+1 T is an n-generator abelian p-group for c k and, in the case p = 2, T/γ c+1 T is a direct product of cyclic groups of exponent > 2; and γ c T Z F and γ c+d T Z F are isomorphic FP-modules for all c k.

48 Proposition (E) Let G be an n-dimensional crystallographic group and let F = F p. We have H i (G/γ c T,F) = H i (G/γ c+d T,F) for all i 0,c k if: P = G/T is a p-group; there exists an integer k 1 such that T/γ c+1 T is an n-generator abelian p-group for c k and, in the case p = 2, T/γ c+1 T is a direct product of cyclic groups of exponent > 2; and γ c T Z F and γ c+d T Z F are isomorphic FP-modules for all c k. So a finite computation can yield the mod-p homology of an infinite family of groups.

49 3. Discrete vector fields A finite regular CW complex is readily stored on a computer

50 A simple homotopy collapse X e n e n 1 ց X e n 1 e n X

51 A simple homotopy collapse X e n e n 1 ց X e n 1 e n X is stored in computer as a pair (e n 1,e n ) or arrow e n 1 e n.

52

53 A discrete vector field is a collection of arrows e n 1 e n with e n 1 in the boundary of e n and with any cell involved in at most one arrow. It is admissible if there is no chain (e n 1 1,e n 1 ),(en 1 2,e n 2 ),(en 1 3,e n 3 ), with each e n 1 i+1 in the boundary of en i and with infinitely many (not necessarily distinct) terms to the right.

54 A regular CW complex X with admissible discrete vector field represents a homotopy equivalence X Y where Y is a CW complex whose cells correspond to those of X that are neither source nor target of any arrow.

55 A regular CW complex X with admissible discrete vector field represents a homotopy equivalence X Y where Y is a CW complex whose cells correspond to those of X that are neither source nor target of any arrow. Such cells of X are critical.

56 A regular CW complex X with admissible discrete vector field represents a homotopy equivalence X Y where Y is a CW complex whose cells correspond to those of X that are neither source nor target of any arrow. Such cells of X are critical. An admissible discrete vector field on X 2 with just one critical 0-cell represents a presentation for π 1 X.

57 A G-CW complex is a CW complex X with an action of a group G that permutes cells. A computer representation is straighforward if X is regular and has only finitely many orbits of cells.

58 A G-CW complex is a CW complex X with an action of a group G that permutes cells. A computer representation is straighforward if X is regular and has only finitely many orbits of cells. C (X) is an exact chain complex of free ZG-modules

59 A G-CW complex is a CW complex X with an action of a group G that permutes cells. A computer representation is straighforward if X is regular and has only finitely many orbits of cells. C (X) is an exact chain complex of free ZG-modules and moreover a resolution with homotopy h : C (X) C +1 (X) dh+hd = 1 if G acts freely and X has a discrete vector field with just one critical cell.

60 A G-CW complex is a CW complex X with an action of a group G that permutes cells. A computer representation is straighforward if X is regular and has only finitely many orbits of cells. C (X) is an exact chain complex of free ZG-modules and moreover a resolution with homotopy h : C (X) C +1 (X) dh+hd = 1 if G acts freely and X has a discrete vector field with just one critical cell. C (Nerve(G)) C (X/G)

61 G = a,b aba = b a: R 2 R 2,(x,y) (x +1, y) b: R 2 R 2,(x,y) (x,y +1) X = R 2 X/G C (Nerve(G)) C (X/G)

62 G = a,b aba = b a: R 2 R 2,(x,y) (x +1, y) b: R 2 R 2,(x,y) (x,y +1) X = R 2 X/G C (Nerve(G)) C (X/G)

63 4. Perturbations and homology of 2-types Joint with Le Van Luyen

64 Theorem (JHC Whitehead): There is an exact sequence of groups π 2 (Y) π 2 (X) π 2 (X,Y) in which is a crossed module: π 1 (Y) π 1 (X) A crossed module is a group homomorphism : M G with action (g,m) g m statisfying ( g m) = g (m)g 1 m m = mm m 1

65 Theorem (JHC Whitehead): There is an exact sequence of groups π 2 (Y) π 2 (X) π 2 (X,Y) in which is a crossed module: π 1 (Y) π 1 (X) A crossed module is a group homomorphism : M G with action (g,m) g m statisfying ( g m) = g (m)g 1 m m = mm m 1 We define π 1 ( ) = G/image π 2 ( ) = ker.

66 Taking Y = X 1 we get Whitehead s functor Ho(CW spaces) Σ 1 (crossed modules) which is faithful on homotopy types X with π n X = 0 for n 1,2. Σ 1 is localization with respect to quasi-isomorphisms

67 Taking Y = X 1 we get Whitehead s functor Ho(CW spaces) Σ 1 (crossed modules) which is faithful on homotopy types X with π n X = 0 for n 1,2. Σ 1 is localization with respect to quasi-isomorphisms Let B(M G) denote a CW-space with π n X = 0 for n 1,2 that maps to.

68 : Q Aut(Q),g {x gxg 1 } is a crossed module for any group Q.

69 : Q Aut(Q),g {x gxg 1 } is a crossed module for any group Q. H 5 ( B(D 216 Aut(D 216 )), Z) = (Z2 ) 11 (Z 3 ) 2 gap> Q:=DihedralGroup(216);; gap> C:=AutoCrossedModule(Q);; gap> Homology(C,5); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 18 ]

70 N : (crossed modules) (simplicial groups) Given : M G we consider the category A = M G s: A A, (m,g) (1,g) t: A A, (m,g) (1, (m)g) : A G A A, ((m,g),(m,g ) (m,( m) 1 g )

71 N : (crossed modules) (simplicial groups) Given : M G we consider the category A = M G s: A A, (m,g) (1,g) t: A A, (m,g) (1, (m)g) : A G A A, ((m,g),(m,g ) (m,( m) 1 g ) s, t, are group homomorphisms and A is a category internal to the category of groups. The nerve N(A) is thus a simplicial group.

72 B: (crossed modules) N (simplicial groups) N (bisimplicial sets) (simplicial sets)

73 B: (crossed modules) N (simplicial groups) F: (sets) (free abelian groups) N (bisimplicial sets) (simplicial sets)

74 B: (crossed modules) N (simplicial groups) F: (sets) (free abelian groups) N (bisimplicial sets) (simplicial sets) C (B( : M G)) is the total complex of the bicomplex: FN 2 N 2 (A) FN 2 N 1 (A) FN 2 N 0 (A) FN 1 N 2 (A) FN 1 N 1 (A) FN 1 N 0 (A) FN 0 N 2 (A) FN 0 N 1 (A) FN 0 N 0 (A)

75 The jth column FN (N j (A)) is the bar complex for the group N j (A).

76 The jth column FN (N j (A)) is the bar complex for the group N j (A). We could replace each column by R N j(a) ZNj (A) Z where R N j(a) is an arbitrary free ZN j (A)-resolution of Z.

77 The jth column FN (N j (A)) is the bar complex for the group N j (A). We could replace each column by R N j(a) ZNj (A) Z where R N j(a) is an arbitrary free ZN j (A)-resolution of Z. But the horizontally induced maps won t square to zero if the resolutions aren t functorial.

78 The jth column FN (N j (A)) is the bar complex for the group N j (A). We could replace each column by R N j(a) ZNj (A) Z where R N j(a) is an arbitrary free ZN j (A)-resolution of Z. But the horizontally induced maps won t square to zero if the resolutions aren t functorial. Homological Perturbation Lemma solves this problem by providing a filtered complex ZN (A) Z R N (A)

79 A homotopy equivalence data (L,d) p i (M,d), h ( ) consists of chain complexes L,M, quasi-isomorphisms i,p and a homotopy ip 1 = dh+hd. A perturbation on ( ) is a homomorphism ǫ: M M of degree 1 such that (d +ǫ) 2 = 0.

80 A homotopy equivalence data (L,d) p i (M,d), h ( ) consists of chain complexes L,M, quasi-isomorphisms i,p and a homotopy ip 1 = dh+hd. A perturbation on ( ) is a homomorphism ǫ: M M of degree 1 such that (d +ǫ) 2 = 0. Perturbation Lemma (M. Crainic 2004): If A = (1 ǫh) 1 ǫ exists then p (L,d i ) (M,d +ǫ), h ( ) is a homotopy equivalence data where i = i +hai, p = p +pah, h = h+hah, d = d +pai.

81 0 0 FN 2 N 2 (A) 0 FN 2 N 1 (A) FN 2 N 0 (A) 0 FN 1 N 2 (A) 0 FN 1 N 1 (A) 0 FN 1 N 0 (A) (M,d) 0 FN 0 N 2 (A) 0 FN 0 N 1 (A) 0 0 FN 0 N 0 (A) 0 R N 2(A) 2 Z 0 R N 1(A) 0 2 Z R N 0(A) 2 Z 0 R N 2(A) 1 Z 0 R N 1(A) 1 Z 0 R N 0(A) 1 Z (L,d) 0 N 2 (A) 0 N 1 (A) 0 N 0 (A)

82 FN 2 N 2 (A) FN 2 N 1 (A) FN 2 N 0 (A) FN 1 N 2 (A) FN 1 N 1 (A) FN 1 N 0 (A) (M,d +ǫ) FN 0 N 2 (A) FN 0 N 1 (A) FN 0 N 0 (A) R N 2(A) 2 Z R N 1(A) 2 Z R N 0(A) 2 Z R N 2(A) 1 Z R N 1(A) 1 Z R N 0(A) 1 Z N 2 (A) N 1 (A) N 0 (A) (L,d )

83 A morphism of crossed modules induces a group homomorphism on π 1 = coker( ) (1) and a π 1 -module homomorphism on π 2 = ker( ). (2) Quasi-isomorphsm is the equivalence relation generated by morphisms where (1) and (2) are isomorphisms. So is quasi-isomorphic to if there exists some sequence of quasi-isomorphisms k

84 : Q Aut(Q) gap> C:=AutoCrossedModule(DihedralGroup(216));; gap> Size(C); # = Q x Aut(Q) gap> IdQuasiCrossedModule(C); [ 72, 68 ]

85 : Q Aut(Q) gap> C:=AutoCrossedModule(DihedralGroup(216));; gap> Size(C); # = Q x Aut(Q) gap> IdQuasiCrossedModule(C); [ 72, 68 ] gap> D:=SmallQuasiCrossedModule(72,68); Crossed module

86 : Q Aut(Q) gap> C:=AutoCrossedModule(DihedralGroup(216));; gap> Size(C); # = Q x Aut(Q) gap> IdQuasiCrossedModule(C); [ 72, 68 ] gap> D:=SmallQuasiCrossedModule(72,68); Crossed module gap> Homology(D,5); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 18 ]

87 A curiosity The crossed module : Z 2 0 yields a homotopy 2-type B = B( ) with π 2 (B) = Z 2, π k (B) = 0 for k 2.

88 A curiosity The crossed module : Z 2 0 yields a homotopy 2-type B = B( ) with π 2 (B) = Z 2, π k (B) = 0 for k 2. gap> B:=EilenbergMacLaneComplex(CyclicGroup(2),2,11);; gap> C:=ChainComplex(B);; gap> List([0..11],CK!.dimension); [ 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 ]

89 A curiosity The crossed module : Z 2 0 yields a homotopy 2-type B = B( ) with π 2 (B) = Z 2, π k (B) = 0 for k 2. gap> B:=EilenbergMacLaneComplex(CyclicGroup(2),2,11);; gap> C:=ChainComplex(B);; gap> List([0..11],CK!.dimension); [ 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 ] gap> D:=CoreducedChainComplex(C);; gap> List([0..10],D!.dimension); [ 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ]