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A Concrete Approach to Abstract Algebra by W. W. Sawyer

By W. W. Sawyer

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Serre, [Se1]. 5 1. Let p = 2, and suppose that I is an admissible monomial for A(2), then the excess of I, e(I), is i 1 - i2 - ... /2) is the polynomial algebra on generators SqI (/"n) where I runs over all monomials of excess less than n. 2. Let p be an odd prime and suppose that I is an admissible monomial for A(p), then the excess, e(I), is i 1 - (p - 1)(i2 + ... + iT) - L:~ Ej. In these terms H*(B zjp ; Z/p) is the tensor product of a polynomial algebra Z/p[·· . ,pI (in),"'] when dim (pI (in)) is even, and and exterior algebra E(-··, pI (in),"') when dim(pI (/"n)) is odd, as I runs over all admissible monomials with e(I) < n.

Then we have a. tr: Hj (H; IFp)-tHj (G;rlp) is the zero map for j even and the identity when j is odd. b. res*: Hj (G; IFp)-tHj (H; IFp) is zero when j is odd and the identity when j is even. Proof. We have LTjb2i(T-jel) = p, j=O p-l = LTje2i+l(T-jel) = 1. res· . tr . O---+Cc ---+C H---+Cc---+O as long as the action of G on the coefficients A is trivial. 10 15. res·. tr. 5. res· ... ---+ H~( G; A) ---+ H~(H; A) ---+ H~(G; A) ---+ H~+l(G; A) ---+ '" . BTl' xid Be ---+ Be x Be ----+ BZ/2 x Be . To prove (b) note that the inclusions H C G induces the chain map of minimal complexes + T + ...

5. Restriction and Transfer In order to determine the cup product a U X in cohomology, we choose representative co chains a for a, c for X and evaluate (a 0 b, L1(w)) as w runs over chains representing all the homology classes in Hdim(o:)+dim(;:() (Bz/p; lF p). This determines the image of aUfJ E Hom(H*(Bz/p;lFp),lFp), and since H*(Bz/p; lF p) = Hom(H*(B z / p; lFp), lF p) this evaluation determines the entire cup product. For example choose a representative, a, for the generator of H2t(Bz/p; lFp) and c for a generator of H2j(Bz/p;lFp).

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