Homological Algebra - Bangalore, November 2016

(1)

Computational Group Cohomology

Bangalore, November 2016

Graham Ellis NUI Galway, Ireland

Slides available at http://hamilton.nuigalway.ie/Bangalore Password: Galway

(2)

Problem

For our favourite group G how do we compute things like:

(3)

Problem

• a resolution

−→R_n^G −→R_n^G₋₁−→ · · · −→R₁^G −→R₀^G ։^η Z of projective ZG-modules

(4)

Problem

• a resolution

• cohomology groups

Hⁿ(G,A) =Hⁿ(HomZG(R∗^G,A))

(5)

Problem

• a resolution

• the Steenrod algebra

H^∗(G,Z₂) =

∞

M

n=1

Hⁿ(G,Z₂)

(6)

Problem

• a resolution

• the Steenrod algebra

H^∗(G,Z₂) =

∞

M

n=1

Hⁿ(G,Z₂)

• Hecke operators, Bredon homology, Tate cohomology ...

(7)

Typical solution in this course

• depends on the type of group G

(8)

• constructs a contractible cellular space X with free G-action and sets

R_∗^G =C∗(X)

(9)

• constructs a contractible cellular space X with free G-action and sets

R_∗^G =C∗(X)

• uses information about an explicit contracting homotopy X ≃ ∗

when computing cohomology groups, Steenrod algebras etc.

(10)

Outline

(11)

Outline

• Lecture 1: CW spaces and their (co)homology

(12)

Outline

• Lecture 2: Algorithms for classifying spaces of groups

(13)

Outline

• Lecture 3: Homotopy 2-types

(14)

Outline

• Lecture 4: Steenrod algebras of finite 2-groups

(15)

Outline

• Lecture 4: Steenrod algebras of finite 2-groups

• Lecture 5: Curvature and classifying spaces of groups

(16)

JHC Whitehead

CW spaces

Simple homotopy theory Crossed modules

· · ·

(17)

CW space

Hausdorff space X with open subsetse_λ ⊂X for λ∈Λ

(18)

CW space

• eλ homeomorphic to unit disk Dⁿ⊂Rⁿ,n = dim(eλ)

(19)

CW space

• X =S

λ∈Λe_λ

(20)

CW space

• X =S

λ∈Λe_λ

• e_λ∩e_λ′ =∅ ifλ6=λ^′

(21)

CW space

• X =S

λ∈Λe_λ

• e_λ∩e_λ′ =∅ ifλ6=λ^′

• φλ:Dⁿ→X

(22)

CW space

• X =S

λ∈Λe_λ

• e_λ∩e_λ′ =∅ ifλ6=λ^′

• φλ:Dⁿ→X

• maps ˚Dⁿ homeomorphically

(23)

CW space

• X =S

λ∈Λe_λ

• e_λ∩e_λ′ =∅ ifλ6=λ^′

• φλ:Dⁿ→X

• mapsSⁿ⁻¹→Xⁿ⁻¹=S

dim(eµ)≤n−1e_µ

(24)

CW space

• X =S

λ∈Λe_λ

• e_λ∩e_λ′ =∅ ifλ6=λ^′

• φλ:Dⁿ→X

• mapsSⁿ⁻¹→Xⁿ⁻¹=S

dim(eµ)≤n−1e_µ

• some extra topological conditions if Λ is infinite

(25)

Regular CW space

Attaching maps φ_λ:Dⁿ→e_λ are homeomorphisms in this case.

regular not regular

(26)

Regular CW space

regular not regular

Represented in GAP as a component object X:

• X!.boundaries[n+1][k] is a list of integers [t,a₁, ...,at] recording that the a_ith cell of dimensionn−1 lies in the boundary of thekth cell of dimensionn.

(27)

Regular CW space

regular not regular

Represented in GAP as a component object X:

• X!.boundaries[n+1][k] is a list of integers [t,a₁, ...,at] recording that the a_ith cell of dimensionn−1 lies in the boundary of thekth cell of dimensionn.

• X!.coboundaries[n][k] is a list of integers [t,a₁, ...,a_t]

recording that the kth cell of dimension n lies in the boundary of the a_ith cell of dimensionn+ 1.

(28)

Motivating problem from applied topology Given a set S of points randomly sampled from an unknown manifold M, what can we infer about the topology ofM?

(29)

Motivating problem from applied topology Given a set S of points randomly sampled from an unknown manifold M, what can we infer about the topology ofM? For instance, S ⊂M ⊂E².

(30)

One approach to the problem

Repeatedly “thicken” the set S to produce a sequence of inclusions S =S₁⊂S₂⊂S₃⊂ · · ·

and then search for “persistent” topological features in the sequence.

(31)

S₁

(32)

S₁ S₂

(33)

S₁ S₂ S₃ S₄

S₅ S₆ S₇ S₈

(34)

Betti numbers β0(X) = number of path components ofX β₁(X) = number of ”1-dimensional holes” in X

(35)

S₁ S₂ S₃ S₄ S₅ S₆ S₇ S₈

β₀ 478 32 9 2 1 1 1 1

β₁ 115 19

These numbers are consistent with the sample coming from some region with the homotopy type of a circle.

(36)

S₁ S₂ S₃ S₄ S₅ S₆ S₇ S₈

β₀ 478 32 9 2 1 1 1 1

β₁ 0 115 18 4 1 1 1 1

(37)

S₁ S₂ S₃ S₄ S₅ S₆ S₇ S₈

β₀ 478 32 9 2 1 1 1 1

β₁ 0 115 18 4 1 1 1 1

These numbers are consistent with the sample coming from some region with the homotopy type of a circle.

(38)

During an inclusion S_s ֒→S_t holes can

persist ֒→

die ֒→

be born ֒→

(39)

persist ֒→

die ֒→

be born ֒→

β_n^st = number ofn-dimensional holes inS_s that persist to S_t

(40)

persist ֒→

die ֒→

be born ֒→

β_n^st = number ofn-dimensional holes inS_s that persist to S_t β₀^st = number of path components inS_s that persist toS_t

(41)

A matrix

(β_n^{s t}) =





1 1 1 0 4 2 0 0 4





can be represented by a βn bar code with

β_n^s,t horizontal lines from column s to column t

(42)

A matrix

(β_n^{s t}) =





1 1 1 0 4 2 0 0 4





can be represented by a βn bar code with

β_n^s,t horizontal lines from column s to column t

(43)

β₁ bar code for our example

(44)

β₀ bar code for our example (first column cropped)

(45)

A B C

D E F

G β₀ bar codes could be enhanced to dendrograms

A B C D E F G

(46)

Second applied topology toy example

F:R²→R³,(x,y)7→(r,g,b)

(47)

F:R²→R³,(x,y)7→(r,g,b) Xt={v ∈R² :Fr(v) +Fg(v) +F_b(v)≤t}

(48)

β₀(Xt) = number of connected components of X_t

(49)

0 t

1 β0

(50)

0 t

1 β0

= rankH₀(X_t,Z)

(51)

Homology functors (degree 1) X_s ֒→X_t

for s ≤t.

(52)

Homology functors (degree 1) X_s ֒→X_t induces

ι^s,t₀ :H₀(X_s,Z)−→H₀(X_t,Z) for s ≤t.

(53)

β₀^s,t(X∗) = rank( image(ι^s,t₀ ) )

(54)

β₀^s,t(X∗) = rank( image(ι^s,t₀ ) )

(55)

ι^s,t₁ :H₁(X_s,Z)−→H₁(X_t,Z) for s ≤t.

β₁^s,t(X∗) = rank( image(ι^s,t₁ ) )

(56)

Chain complex of a regular CW space X C_nX = free abelian group on the set of n-cells ofX

(57)

∂n:C_nX →C_n−₁X,e_λⁿ7→X

ǫλ,µe_µⁿ⁻¹

ǫλ,µ=

±1 if e_µⁿ⁻¹ lies in closure of e_λⁿ 0 otherwise

(58)

ǫλ,µe_µⁿ⁻¹

ǫλ,µ=

Sign of ǫλ,µ chosen (non-uniquely) to ensure

∂n∂n+1 = 0

(59)

ǫλ,µe_µⁿ⁻¹

ǫλ,µ=

Sign of ǫλ,µ chosen (non-uniquely) to ensure

∂n∂n+1 = 0

Homology of a regular CW space X

H_n(X,Z) =H_n(C∗X) = ker∂n/image∂_n+1

(60)

Example

X =

C₃X ^∂³ ^//C₂X ^∂² ^//C₁X ^∂¹ ^//C₀X

0 ^//Z²⁴⁹⁷⁰² ^//Z⁵⁰⁹⁴¹⁷ ^//Z²⁵⁹⁶⁰¹

(61)

Example

X =

C₃X ^∂³ ^//C₂X ^∂² ^//C₁X ^∂¹ ^//C₀X

0 ^//Z²⁴⁹⁷⁰² ^//Z⁵⁰⁹⁴¹⁷ ^//Z²⁵⁹⁶⁰¹ The matrix of ∂₁ has 132245162617 entries.

Direct application of Smith Normal Form is not so practical in such a situation.

(62)

A simple homotopy collapse Write

Y ցX if

Y =X ∪eⁿ∪eⁿ⁻¹

X

eⁿ

eⁿ⁻¹

(63)

A simple homotopy collapse Write

Y ցX if

Y =X ∪eⁿ∪eⁿ⁻¹

X

eⁿ

eⁿ⁻¹

Such a collapse is recorded as an arrow eⁿ⁻¹ −→eⁿ.

(64)

Example The homotopy equivalence [0,4]×[0,4]≃ ∗

≃ ∗

can be realized as a collection of simple homotopy collapses

(65)

Bing’s house

(66)

Bing’s house

(67)

Bing’s house

The homotopy equivalence

Bing^′s house≃ ∗

is not representable as a collection of simple homotopy collapses.

(68)

A discrete vector field

is a collection of arrows eⁿ⁻¹ −→eⁿ witheⁿ⁻¹ in the boundary of eⁿ and with any cell involved in at most one arrow. It is

admissible if there is no chain

· · ·(e₁ⁿ⁻¹ →e₁ⁿ),(e₂ⁿ⁻¹ →e₂ⁿ),(e₃ⁿ⁻¹ →e₃ⁿ),· · ·

with eache_iⁿ⁻¹₊₁ in the boundary ofe_iⁿ and with infinitely many (not necessarily distinct) terms to the right.

1 1

2 2

3 3

4 4

1 1

5 6 7

(69)

is a collection of arrows eⁿ⁻¹ −→eⁿ witheⁿ⁻¹ in the boundary of eⁿ and with any cell involved in at most one arrow.

1 1

2 2

3 3

4 4

1 1

5 6 7

(70)

is a collection of arrows eⁿ⁻¹ −→eⁿ witheⁿ⁻¹ in the boundary of eⁿ and with any cell involved in at most one arrow. It is

admissible if there is no chain

· · ·(e₁ⁿ⁻¹ →e₁ⁿ),(e₂ⁿ⁻¹ →e₂ⁿ),(e₃ⁿ⁻¹ →e₃ⁿ),· · ·

with eache_iⁿ⁻¹₊₁ in the boundary ofe_iⁿ and with infinitely many (not necessarily distinct) terms to the right.

1 1

2 2

3 3

4 4

1 1

5 6 7

(71)

Theorem

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 ofX that are neither source nor target of any arrow.

1 1

2 2

3 3

4 4

1 1

5 6 7

(72)

Theorem

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 ofX that are neither source nor target of any arrow. Such cells of X are critical.

1 1

2 2

3 3

4 4

1 1

5 6 7

(73)

Second toy example revisited

X = ≃X^′

C₃X^′ ^∂³ ^//C₂X^′ ^∂² ^//C₁X^′ ^∂¹ ^//C₀X^′

0 ^//Z⁰ ^//Z⁴⁷⁶ ^//Z³⁶²

(74)

gap> M:=ReadImageAsPureCubicalComplex("file.png",300);

Pure cubical complex of dimension 2.

(75)

gap> Y:=RegularCWComplex(M);

Regular CW-complex of dimension 2

(76)

gap> C:=ChainComplexOfRegularCWComplex(Y);

Chain complex of length 2 in characteristic 0 . gap> List([0..2],C!.dimension);

[ 259601, 509417, 249702 ]

(77)

gap> C:=ChainComplexOfRegularCWComplex(Y);

Chain complex of length 2 in characteristic 0 . gap> List([0..2],C!.dimension);

[ 259601, 509417, 249702 ] gap> MM:=ContractedComplex(M);;

gap> YY:=RegularCWComplex(MM);;

gap> YY:=ContractedComplex(YY);;

gap> CC:=ChainComplex(YY);;

gap> List([0..2],CC!.dimension);

[ 362, 476, 0 ]

(78)

Chain equivalence

Let X be a regular CW-space with discrete vector field. LetD_n denote the subgroup of C_nX freely generated by critical n-cells.

There are commuting homomorphisms C_nX ^φⁿ ^//

∂n

D_n

δn

C_n−1X^φⁿ⁻¹ ^//D_n

C_nX

∂n

D_n

ψn

oo

δn

C_n−1X^ψ^ooⁿ⁻¹ D_n satisfying

∂_n∂_n+1= 0, δ_nδ_n+1= 0

(79)

Chain equivalence

Let X be a regular CW-space with discrete vector field. LetD_n denote the subgroup of C_nX freely generated by critical n-cells.

There are commuting homomorphisms C_nX ^φⁿ ^//

∂n

D_n

δn

C_n−1X^φⁿ⁻¹ ^//D_n

C_nX

∂n

D_n

ψn

oo

δn

C_n−1X^ψ^ooⁿ⁻¹ D_n satisfying

∂_n∂_n+1= 0, δ_nδ_n+1= 0 and inducing isomorphisms

Hn(φ∗) :Hn(C∗X)−→^∼⁼ Hn(D∗) H_n(ψ∗) =H_n(φ∗)⁻¹:H_n(D∗)−→^∼⁼ H_n(C∗X)

(80)

Hⁿ(G,Z) =Hⁿ(HomZG(C∗X,Z))

G group

X contractible CW-space with free G-action

(81)

G group

X contractible CW-space with free G-action Example

G =ha,b|bab=ai, X =R²,

a b

(82)

G group

X contractible CW-space with free G-action Example

G =ha,b|bab=ai, X =R², G \X = Klein bottle

a b

(83)

G group

X contractible CW-space with free G-action

Hⁿ(−,Z) contravariant functorGroups −→Abelian groups Example

G =ha,b|bab=ai, X =R², G \X = Klein bottle

a b

(84)

Contracting homotopy Exactness of the free ZG-resolution

R∗=C∗X : · · · −→R_n−→R_n−1 −→ · · · −→R₀

is encoded as Z-linear homomorphismsh_n:R_n−→R_n+1 satisfying h_n−₁∂n+∂_n+1h_n (n ≥1).

(85)

Contracting homotopy Exactness of the free ZG-resolution

R∗=C∗X : · · · −→R_n−→R_n−1 −→ · · · −→R₀

is encoded as Z-linear homomorphismsh_n:R_n−→R_n+1 satisfying h_n−₁∂n+∂_n+1h_n (n ≥1).

Example continued

h_n can encode a contracting distrete vector field onX

(86)

Classical homological algebra Given

x ∈ker(Rn

∂n

−→Rn−1) we can, by exactness, choose some

˜

x ∈R_n+1 such that

∂_n+1x˜=x

(87)

Classical homological algebra Given

x ∈ker(Rn

∂n

−→Rn−1) we can, by exactness, choose some

˜

x ∈R_n+1 such that

∂_n+1x˜=x

Constructive homological algebra

We can simply set

˜

x=h_n(x)

(88)

A group theoretic example of vector fields

Ap(G) = poset of non-trivial elementary abelian subgroups inG.

∆Ap(G) = order (simplicial) complex of Ap(G)

(89)

∆Ap(G) = order (simplicial) complex of Ap(G) Proposition (Ksontini) ∆A2(S₇) =∨¹⁶⁰_i₌₁S²

(90)

∆Ap(G) = order (simplicial) complex of Ap(G) Proposition (Ksontini) ∆A2(S₇) =∨¹⁶⁰_i₌₁S²

gap> K:=QuillenComplex(SymmetricGroup(7),2);

Simplicial complex of dimension 2.

gap> Size(K);

11291

gap> Y:=RegularCWComplex(K);;

gap> C:=CriticalCells(Y);;

gap> n:=0;;Length(Filtered(C,c->c[1]=n));

1

0

160