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Computations Induced (Co)homology homomorphism

Daher Al Baydli

Department of Mathematics National University of Ireland,Galway

Feberuary, 2017

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 1 / 11 Overview

Resolution Chain and cochain complex homology cohomology Induce (Co)homology homomorphism

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 2 / 11 Resolution

Definition Let G be a group and Z be the group of integers considered as a trivial ZG-module. The map  : ZG −→ Z from the integral group ring to Z, given by Σmg g 7−→ Σmg , is called the augmentation.

Definition Let G be a group. A free ZG-resolution of Z is an exact sequence of free G G ∂n+1 G ∂n G ∂n−1 ∂2 G ∂1 ZG-modules R∗ : · · · −→ Rn+1 −→ Rn −→ Rn−1 −→ · · · −→ R1 −→ G  G R0 −→ R−1 = Z −→ 0 G with each Ri a free ZG-module for all i ≥ 0.

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 3 / 11 Definition

Let C = (Cn, ∂n)n∈Z be a chain complex of Z-modules. For each n ∈ Z, the nth homology module of C is defined to be the quotient module

Ker∂n Hn(C) = Im∂n+1

G We can also construct an induced cochain complex HomZG (R∗ , A) of abelian groups: G G δn G δn−1 HomZG (R∗ , A): · · · ←− HomZG (Rn+1, A) ←− HomZG (Rn , A) ←− G δn−2 δ1 G δ0 G HomZG (Rn−1, A) ←− · · · ←− HomZG (R1 , A) ←− HomZG (R0 , A)

Chain and cochain complex

G Given a ZG-resolution R∗ of Z and any ZG-module A we can use the G tensor product to construct an induced chain complex R∗ ⊗ZG A of G G ∂n+1 G ∂n abelian groups: R∗ ⊗ZG A : · · · −→ Rn+1 ⊗ZG A −→ Rn ⊗ZG A −→ G ∂n−1 ∂2 G ∂1 G Rn−1 ⊗ZG A −→ · · · −→ R1 ⊗ZG A −→ R0 ⊗ZG A

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 4 / 11 Definition

Let C = (Cn, ∂n)n∈Z be a chain complex of Z-modules. For each n ∈ Z, the nth homology module of C is defined to be the quotient module

Ker∂n Hn(C) = Im∂n+1

Chain and cochain complex

G Given a ZG-resolution R∗ of Z and any ZG-module A we can use the G tensor product to construct an induced chain complex R∗ ⊗ZG A of G G ∂n+1 G ∂n abelian groups: R∗ ⊗ZG A : · · · −→ Rn+1 ⊗ZG A −→ Rn ⊗ZG A −→ G ∂n−1 ∂2 G ∂1 G Rn−1 ⊗ZG A −→ · · · −→ R1 ⊗ZG A −→ R0 ⊗ZG A G We can also construct an induced cochain complex HomZG (R∗ , A) of abelian groups: G G δn G δn−1 HomZG (R∗ , A): · · · ←− HomZG (Rn+1, A) ←− HomZG (Rn , A) ←− G δn−2 δ1 G δ0 G HomZG (Rn−1, A) ←− · · · ←− HomZG (R1 , A) ←− HomZG (R0 , A)

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 4 / 11 Chain and cochain complex

G Given a ZG-resolution R∗ of Z and any ZG-module A we can use the G tensor product to construct an induced chain complex R∗ ⊗ZG A of G G ∂n+1 G ∂n abelian groups: R∗ ⊗ZG A : · · · −→ Rn+1 ⊗ZG A −→ Rn ⊗ZG A −→ G ∂n−1 ∂2 G ∂1 G Rn−1 ⊗ZG A −→ · · · −→ R1 ⊗ZG A −→ R0 ⊗ZG A G We can also construct an induced cochain complex HomZG (R∗ , A) of abelian groups: G G δn G δn−1 HomZG (R∗ , A): · · · ←− HomZG (Rn+1, A) ←− HomZG (Rn , A) ←− G δn−2 δ1 G δ0 G HomZG (Rn−1, A) ←− · · · ←− HomZG (R1 , A) ←− HomZG (R0 , A) Definition

Let C = (Cn, ∂n)n∈Z be a chain complex of Z-modules. For each n ∈ Z, the nth homology module of C is defined to be the quotient module

Ker∂n Hn(C) = Im∂n+1

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 4 / 11 Induce (Co)homology homomorphism

Definition ∗ Given a chain complex and a group G, we can define the cochains Cn to be the respective groups of all homomorphisms from Cn to G: ∗ Cn = Hom(Cn, G). We define the coboundary map δn−1 : Cn−1 −→ Cn dual to ∂n. we can define the nth cohomology group as the quotient:

n ∗ kerδn H (Cn ) = . Imδn−1

Let C and D be chain complexes and let f : C −→ D be a chain homomorphism. The commutativity of the diagram

∂ Cn −→ Cn−1 fn ↓ ↓ fn−1 ∂ Dn −→ Dn−1

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 5 / 11 Induce (Co)homology homomorphism

implies that fn restrict to maps from Zn(C) into Zn(D) (where Zn is cycle) and Bn(C) into Bn(D) (where Bn is boundary). Hence it induces a homomorphism, f∗ : Hn(C) −→ Hn(D) Then f∗ is called the induced homomorphism on homology groups.

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 6 / 11 Induce (Co)homology homomorphism

Let f 0 be the dual cochain homomorphism from C ∗ to D∗. As in the case of chain homomorphism, that we have a commutative diagram

C n −→δ C n+1 f n ↓ ↓ f n+1 Dn −→δ Dn+1

which implies f 0n restricts to maps from Z n(C) and Bn(C) into Z n(D) and Bn(D) (where Z n is cocycle and Bn is coboundary), respectively, and so induces a homomorphism f ∗ : Hn(C) −→ Hn(D) Then f ∗ is called the induced homomorphism on cohomology groups.

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 7 / 11 Induce (Co)homology homomorphism

Theorem Let R∗ be a free ZG-resolution of Z. Let A be a ZG-module. There is an isomorphism ∼ rn Rn ⊗ G A = A × A × · · · × A = A , Z | {z } rn where rn = rankZG (Rn).

Example Consider the symmetric group G = S4 and the identity map φ : Z4 −→ Z4. The following commands then compute the homology homomorphism

H3(φ) H3(S4, Z/4Z) −→ H3(S4, Z/4Z)

and determine that the kernel of this homomorphism has Size 1.

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 8 / 11 Example

GAP session gap > G := SymmetricGroup(4); ; gap > a := AbelianGroup([4]); ; gap > b := a;; gap > ahomb := GroupHomomorphismByFunction(a, b, x− > x); ; gap > A := TrivialGModuleAsGOuterGroup(G, a); ; gap > B := TrivialGModuleAsGOuterGroup(G, b); ; gap > phi := GOuterGroupHomomorphism(); ; gap > phi!.Source := A;; gap > phi!.Target := B;; gap > phi!.Mapping := ahomb;; gap > Hphi := HomologyHomomorphism(phi, 3); ; gap > Size(KernelOfGOuterGroupHomomorphism(Hphi)); 1

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 9 / 11 Summary

We devise and implement an algorithm for computing the induced homology homomorphism Hn(φ): Hn(G, A) −→ Hn(G, B). We devise and implement an algorithm for computing induced cohomology homomorphism Hn(φ): Hn(G, A) −→ Hn(G, B).

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 10 / 11 BibliographyI

1-Hatcher, Allen, Algebraic topology, Cambridge University Press, New York, NY, USA.

2- Brown, Kenneth S., Cohomology of Groups, Graduate Texts in Mathematics, 87, Springer, 1994.

3-Graham Ellis, HAP - Homological Algebra Programming, Version 1.11.3, 2015.

Daher Al Baydli (NUIGalway) Computations Induced (Co)homology homomorphism Feberuary, 2017 11 / 11