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edit approved Mar 16, 2015 at 14:51

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crossede Crossed module classification theorem H^3and $H^3(G; A)$

We are interested in "Crossed Molule Classification Teorem (Two crossed modules with kernel A and cokernel G detemined the same class in H³(G;A) if and only if they are aquivalent.)"
We study the works of Charles A.Weibel "An Introduction To Homological Algebra", Saunders Mac Lane "Historical Note, Cohomolgy Groups  Hⁿ(G;M)" and Joseph J.Rotman " An Intoduction to Homological Algebra"
Weibel says:
Consider a 4-term exact sequence with A central in N; 0→A→N→E→G→1 and choose a based section σ:G→E of π; as in the theory of factor sets, the map []:G×G→kerπ defined by [g,h]=σ(g)σ(h)σ(gh)⁻¹ satisfies a nonabelian cocycle condition [f,g][fg,h]=σ(f)[g,h]σ(f)⁻¹[f,gh]. we can lift each [f,g] to an element [[f,g]] of N. 
Given a crossed module N→E we set A=kerα and G=cokerα; G is a group becouse α(N) is normal in E, A is in the center of N and G acts on A, so that A is a G module and we have the sequence 0→A→N→E→G→1. Assuming that N→E is a crossed module, the failure of [[f,g]] to satisfied the cocycle condition is given by the function c:G×G×G→A define by c(f,g,h)[[f,g]][[fg,h]]=σ(f)[[g,h]]σ(f)⁻¹[[f,gh]]. 
How can we check that c is a 3-cocycle, whose class in H³(G;A) is independent of choices of σ and [[f,g]]?
Can we define the function of c explicitly?

We are interested in "Crossed Module Classification Theorem (Two crossed modules with kernel $A$ and cokernel $G$ determined the same class in $H^3(G;A)$ if and only if they are equivalent.)".

We study the works of Charles A. Weibel "An Introduction To Homological Algebra", Saunders Mac Lane "Historical Note, Cohomolgy Groups $H^n(G;M)$" and Joseph J. Rotman "An Introduction to Homological Algebra". Weibel says:

Consider a $4$-term exact sequence with $A$ central in $N$; $0\to A\to N\to E\to G\to 1$ and choose a based section $\sigma : G \to E$ of $\pi$; as in the theory of factor sets, the map $[] : G\times G \to \ker\pi$ defined by $[g,h] = \sigma(g)\sigma(h)\sigma(gh)^{-1}$ satisfies a nonabelian cocycle condition $[f,g][fg,h] = \sigma(f)[g,h]\sigma(f)^{-1}[f,gh]$. We can lift each $[f,g]$ to an element $[[f,g]]$ of $N$.

Given a crossed module $N\to E$ we set $A = \ker\alpha$ and $G = \operatorname{coker}\alpha$; $G$ is a group because $\alpha(N)$ is normal in $E$, $A$ is in the center of $N$ and $G$ acts on $A$, so that $A$ is a $G$ module and we have the sequence $0\to A\to N\to E\to G\to 1$. Assuming that $N\to E$ is a crossed module, the failure of $[[f,g]]$ to satisfy the cocycle condition is given by the function $c : G\times G\times G \to A$ defined by $c(f,g,h)[[f,g]][[fg,h]] = \sigma(f)[[g,h]]\sigma(f)^{-1}[[f,gh]]$.

How can we check that $c$ is a $3$-cocycle, whose class in $H^3(G;A)$ is independent of choices of $\sigma$ and $[[f,g]]$? Can we define the function of $c$ explicitly?

crossede module classification theorem H^3

We are interested in "Crossed Molule Classification Teorem (Two crossed modules with kernel A and cokernel G detemined the same class in H³(G;A) if and only if they are aquivalent.)"
We study the works of Charles A.Weibel "An Introduction To Homological Algebra", Saunders Mac Lane "Historical Note, Cohomolgy Groups  Hⁿ(G;M)" and Joseph J.Rotman " An Intoduction to Homological Algebra"
Weibel says:
Consider a 4-term exact sequence with A central in N; 0→A→N→E→G→1 and choose a based section σ:G→E of π; as in the theory of factor sets, the map []:G×G→kerπ defined by [g,h]=σ(g)σ(h)σ(gh)⁻¹ satisfies a nonabelian cocycle condition [f,g][fg,h]=σ(f)[g,h]σ(f)⁻¹[f,gh]. we can lift each [f,g] to an element [[f,g]] of N. 
Given a crossed module N→E we set A=kerα and G=cokerα; G is a group becouse α(N) is normal in E, A is in the center of N and G acts on A, so that A is a G module and we have the sequence 0→A→N→E→G→1. Assuming that N→E is a crossed module, the failure of [[f,g]] to satisfied the cocycle condition is given by the function c:G×G×G→A define by c(f,g,h)[[f,g]][[fg,h]]=σ(f)[[g,h]]σ(f)⁻¹[[f,gh]]. 
How can we check that c is a 3-cocycle, whose class in H³(G;A) is independent of choices of σ and [[f,g]]?
Can we define the function of c explicitly?

Crossed module classification theorem and $H^3(G; A)$

We are interested in "Crossed Module Classification Theorem (Two crossed modules with kernel $A$ and cokernel $G$ determined the same class in $H^3(G;A)$ if and only if they are equivalent.)".

Consider a $4$-term exact sequence with $A$ central in $N$; $0\to A\to N\to E\to G\to 1$ and choose a based section $\sigma : G \to E$ of $\pi$; as in the theory of factor sets, the map $[] : G\times G \to \ker\pi$ defined by $[g,h] = \sigma(g)\sigma(h)\sigma(gh)^{-1}$ satisfies a nonabelian cocycle condition $[f,g][fg,h] = \sigma(f)[g,h]\sigma(f)^{-1}[f,gh]$. We can lift each $[f,g]$ to an element $[[f,g]]$ of $N$.

Given a crossed module $N\to E$ we set $A = \ker\alpha$ and $G = \operatorname{coker}\alpha$; $G$ is a group because $\alpha(N)$ is normal in $E$, $A$ is in the center of $N$ and $G$ acts on $A$, so that $A$ is a $G$ module and we have the sequence $0\to A\to N\to E\to G\to 1$. Assuming that $N\to E$ is a crossed module, the failure of $[[f,g]]$ to satisfy the cocycle condition is given by the function $c : G\times G\times G \to A$ defined by $c(f,g,h)[[f,g]][[fg,h]] = \sigma(f)[[g,h]]\sigma(f)^{-1}[[f,gh]]$.

How can we check that $c$ is a $3$-cocycle, whose class in $H^3(G;A)$ is independent of choices of $\sigma$ and $[[f,g]]$? Can we define the function of $c$ explicitly?

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asked Mar 16, 2015 at 13:17

hatice gülsün

crossede module classification theorem H^3

We are interested in "Crossed Molule Classification Teorem (Two crossed modules with kernel A and cokernel G detemined the same class in H³(G;A) if and only if they are aquivalent.)"
We study the works of Charles A.Weibel "An Introduction To Homological Algebra", Saunders Mac Lane "Historical Note, Cohomolgy Groups  Hⁿ(G;M)" and Joseph J.Rotman " An Intoduction to Homological Algebra"
Weibel says:
Consider a 4-term exact sequence with A central in N; 0→A→N→E→G→1 and choose a based section σ:G→E of π; as in the theory of factor sets, the map []:G×G→kerπ defined by [g,h]=σ(g)σ(h)σ(gh)⁻¹ satisfies a nonabelian cocycle condition [f,g][fg,h]=σ(f)[g,h]σ(f)⁻¹[f,gh]. we can lift each [f,g] to an element [[f,g]] of N. 
Given a crossed module N→E we set A=kerα and G=cokerα; G is a group becouse α(N) is normal in E, A is in the center of N and G acts on A, so that A is a G module and we have the sequence 0→A→N→E→G→1. Assuming that N→E is a crossed module, the failure of [[f,g]] to satisfied the cocycle condition is given by the function c:G×G×G→A define by c(f,g,h)[[f,g]][[fg,h]]=σ(f)[[g,h]]σ(f)⁻¹[[f,gh]]. 
How can we check that c is a 3-cocycle, whose class in H³(G;A) is independent of choices of σ and [[f,g]]?
Can we define the function of c explicitly?

cohomology