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Cohomology of Groups by Edwin Weiss

By Edwin Weiss

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C U E ~ O€C = O} ~ = /oSC~ n , ~ EaeCr / ~ n . = /O€C o /~ = ~ 1-5-1. = ~ 1 ( ~ - i ) I. 28 COHOMOLOGY GROUPS OF G IN A Proof: An n-cochain (with respect to the standard complex X = X8) is completely determined by its values on the n-cells. PI -fb,TI d 7 , Vo, T , p E G Thus, f is a cocycle if and only if it satisfies Furthermore, the 2-cochain f is a coboundary if and only if it is of the form f = Sg for some g E %“-thus, it must satisfy Functions of two variables satisfying the cocycle identity were known classically as factor sets, while those satisfying the coboundary identity were known as splitting factor sets.

Now, suppose inductively that A, is defined and that a,~I,a;+~ = 0. Since X;+, is G'-free it is G'regular, so by (2-1-4) there exists T & ~E Hom (XA+l,XA+l) such that S'(n;+l) = 1 (the identity map on X;+,). Put An+, = s(QAak+,rk+l) is a G-homomorphism, and using the fact that G'Thus, homomorphisms slip through the trace S', we have %+Jn+, = S(an+,~n4zaA+i4+1) = f(A%+14+1) - s(Qa-lan@k+14+J (since an+,Dn Dn,an = 1) + = Anak+ls(Tk+J = An%+, 2-1. HOMOMORPHISM OF PAIRS 49 Finally, a,+lA,+laA+2 = A,a&,a&, = 0, so the induction hypothesis is satisfied.

COHOMOLOGY GROUPS OF G IN A 27 In other words, is a differential graded group of cohomology type which may also be written as the sequence : **. * We call V" = Vn(G, A) = Hom, (X, , A) the group of n-cochains of G in A, 2Tn = %""(G, A) = { f E $fn I Sf = 0} the group of n-cocycles of G in A, and an= Bn(G, A) = SVn-l the group of n-coboundaries of G in A. The nth derived group H,(Hom, (X, A)) = P/anis known as the nth cohomology group of C in A; it is denoted by Hn(G,A). Two n-cocycles f and g are said to be cohomologous (denoted f -g) when they differ by a coboundary-that is, when they determine the same element in the cohomology group Hn(G,A).

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