On the Integral Cohomology of Wreath Products

Abstract

Under mild conditions on the space X, we describe the additive structure of the integral cohomology of the space X p = EC in terms of the cohomology of X. C p pWe give weaker results for other similar spaces, and deduce various corollaries concerning the cohomology of finite groups. ᮊ 1997 Academic Press INTRODUCTIONLet S be a group with a fixed action on a finite set ⍀. By the wreath product G X S of a group G with S we mean a split extension with kernel G ⍀ , quotient S, and with the S-action on G ⍀ given by permuting the copies of G. Our main interest is the integral cohomology of finite groups of the form G X S. We work in greater generality, however, because it is no more difficult to study the cohomology of spaces of the formwhere E is an S-free S-CW-complex, and X is a CW-complex of finite type. The mod-p cohomology of certain such spaces plays a crucial role in w x Steenrod's definition of the reduced power operations 29 . Building on work of Steenrod, Nakaoka described the cohomology of such spaces with w x coefficients in any field 23 . The point about working over a field is that then the cellular cochain complex for X is homotopy equivalent to the cohomology of X, viewed as a complex with trivial differential. If the integral cohomology of X is free, then a similar result holds in this case.Ž . Evens used this to study the cohomology of the classifying space of the Ž . Lie group U m X ⌺ in the course of his work on Chern classes of n w x induced representations 13 . The study of the integral cohomology in the U Ž . case when H X is not free is much harder. The pioneers in this case were Evens and Kahn, who made a partial study of the important special 184 0021-8693r97 $25.00Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. COHOMOLOGY OF WREATH PRODUCTS 185case of X p = EC . We complete the study of this case in Section 4 C p p below, which could be viewed as both an extension of and a simplification w x of 15, Sect. 4 . Many, but not all, of our results are corollaries of this work. Our paper has the following structure.In Section 1 we give some algebraic background. Most of this material is well-known, although we have not seen Lemma 1.4 stated explicitly before, and we believe that Lemma 1.1 is original. This lemma, which compares spectral sequences coming from double complexes consisting of ''the same groups,'' but with ''different differentials,'' is the key to our extension of w x the Evens᎐Kahn results 15 . Theorem 2.1 is a statement of the result of Nakaoka mentioned above, which for interest's sake we have made more general than the original. A weak version of this theorem could be stated U Ž . as ''if H X is free, then the Cartan᎐Leray spectral sequence for U Ž ⍀ . H X = E collapses at the E -page.'' A similarly weakened version of S 2 Ž . Theorem 2.2 would say that over the integers or any PID , this spectral < < sequence collapses at the E -page, r s 2 q ⍀ , without any condition onThe Cartan᎐Leray spectral sequence may be obtained from a double complex. In...

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