For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

We show that the Morava K -theories of the groups of order 32 are concentrated in even degrees. 55N20, 55R35; 57T25Let G be a finite group and BG denote its classifying space. Determining the Morava K -theory of BG is generally difficult, mainly due to the weakness of existing methods of calculation, which all require knowledge of the cohomology of p -groups -in itself a notorious problem. Certain series of groups with particularly simple structure, such as wreath products, groups having a cyclic maximal subgroup or minimal nonabelian groups, are quite tractable; see eg the work of Hopkins, Kuhn In this note we shall consider the groups of order 32. In many cases the Morava K -theory is already known, or easily deduced from results in the literature. For the remaining groups, the author established in [7] that for n D 2 at least, their Morava Ktheory K.n/ .BG/ is generated by transfers of Euler classes of complex representations. In other words, all groups of order 32 are "K.2/-good" in the sense of Hopkins, Kuhn and Ravenel. Some of the results however relied on computer calculations. This is to be remedied here, although we only prove a weaker statement:Theorem Let G be a group of order 32. Then K.n/ odd .BG/ D 0.When starting this project, our objective of course was not to prove such a result, rather we hoped -rather naively, perhaps -that order 32 would be big enough to find a 2-primary counterexample to the even degree conjecture. In this we have failed and the problem remains open.

Abstract. We compute the Morava K-theory of some extraspecial 2-groups and associated compact groups.

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Each permutation representation of a finite group G can be used to pull cohomology classes back from a symmetric group to G. We study the ring generated by all classes that arise in this fashion, describing its variety in terms of the subgroup structure of G.We also investigate the effect of restricting to special types of permutation representations, such as GLn(Fp) acting on flags of subspaces.

Abstract. The BP * -module structure of BP * (BG) for extraspecial 2-groups is studied using transfer and Chern classes. These give rise to p-torsion elements in the kernel of the cycle map from the Chow ring to ordinary cohomology first obtained by Totaro.

We study natural subalgebras Ch E (BG; R) of group cohomology H * (BG; R) defined in terms of the infinite loop spaces in spectra E and give representation theoretic descriptions of those based on QS 0 and the Johnson-Wilson theories E(n). We describe the subalgebras arising from the Brown-Peterson spectra BP and as a result give a simple reproof of Yagita's theorem that the image of BP * (BG) in H * (BG; F p ) is F-isomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a non-trivial map of ring spectra E → H F p . We also extend our constructions to define subalgebras of H * (X; R) for any space X; when X is a finite CW complex we show that the subalgebras Ch E(n) (X; R) give a natural unstable chromatic filtration of H * (X; R). ᭧

We show that K(2)-locally, the smash product of the string bordism spectrum and the spectrum T 2 splits into copies of Morava E-theories.Here, T 2 is related to the Thom spectrum of the canonical bundle over ΩSU (4).