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Homology Homomorphism || 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)n2Z be a chain complex of Z-modules. For each n 2 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)n2Z be a chain complex of Z-modules. For each n 2 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)n2Z be a chain complex of Z-modules. For each n 2 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.
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