Equivariant homology decompositions

Kahn, Peter J.

doi:10.1090/S0002-9947-1986-0857444-0

Equivariant homology decompositions
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Trans. Amer. Math. Soc. 298 (1986), 273-287 Request permission

Abstract:

This paper presents some results on the existence of homology decompositions in the context of the equivariant homotopy theory of Bredon. To avoid certain obstructions to the existence of equivariant Moore spaces occurring already in classical equivariant homotopy theory, most of the work of this paper is done “over the rationals.” The standard construction of homology decompositions by Eckmann and Hilton can be followed in the present equivariant context until it is necessary to produce appropriate $k’$-invariants. For these, the Eckmann-Hilton construction uses a certain Universal Coefficient Theorem for homotopy sets. The relevant extension of this to the equivariant situation is an equivariant Federer spectral sequence, which is developed in $\S 2$. Using this, we can formulate conditions which imply the existence of the desired $k’$-invariants, and hence the existence of the homology decomposition. The conditions involve a certain notion of projective dimension. For one application, equivariant homology decompositions always exist when the group has prime order.

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