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By J. L. Dupont, I. H. Madsen

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Davis Many mathematicians have worked on related ideas; we refer to [36,55,170], and [56] for references to original sources. These ideas were pioneered by Jones [99] and were taken farthest by S. Weinberger and his collaborators. For the rest of this section, let q denote the order of the finite group G. 1. A G-action on Y propagates across a map / : X -> y if there exists a G-action on X and an equivariant map homotopic to / . DEFINITION Similarly given a G-action on X one can talk about propagation across a map / : X ^- Y.

7. Acyclic complexes and the Conner conjecture If Z is a G space and / / C G is a subgroup then a basic construction is the transfer map C* {X/ G) -> C* (X/H). By averaging on cochains it is elementary to construct such a map (see [33]) with the property that composed with the projection X/H -^ X/G the resulting map is multipHcation by [G : //] on H'^iX/G, A), where A is any coefficient group. Note in particular that if P = Syl^(G), we have an embedding //*(X/G; F^) -> H*(X/P; ¥p). 15. IfXis a finite-dimensional acyclic G-complex, then X/G is acyclic.

More generally it is possible to use this approach to construct actions of arbitrary 2-groups on products of spheres with maximal isotropy of rank equal to the co-rank in G of the largest central elementary abelian subgroup. How to build a free action on a larger product from this object is still unknown. In the context of representation theory the work of U. Ray (see [145]) is also relevant here. She proves that if G is a finite group acting freely on a product of spheres arising from G-representations, then the only possible non-abelian composition factors of G are the alternating groups A5 and A(y.

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