On the cohomology of split extensions of finite groups

Title: On the cohomology of split extensions of finite groups

Author: Stephen F. Siegel

This paper will appear in the Trans. Amer. Math. Soc.

Abstract: Let $G=H\rtimes Q$ be a split extension of finite groups,
and consider the Lyndon-Hochschild-Serre spectral sequence of the
extension with coefficients in a field $k$.  The $E_{2}$-page is
isomorphic to $H^*(Q,H^*(H,k))$ and it converges to $H^*(G,k)$ modulo
a certain filtration.  A theorem of Charlap and Vasquez gives an
explicit description of the differentials $d_2$ in this case.  We
generalize this to give an explicit description of all the $d_r$
($r\geq 2$).  The generalization is obtained by associating to the
group extension a certain twisting cochain.  The twisting cochain not
only determines the differentials, but also allows one to construct an
explicit $kG$-projective resolution of $k$.

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Author address:
        Stephen F. Siegel
        Department of Mathematics
        Northwestern University
        Evanston, IL 60208-2730
        708-491-5594