Volume 11, issue 3 (2011)

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Units of equivariant ring spectra

Rekha Santhanam

Algebraic & Geometric Topology 11 (2011) 1361–1403
Abstract

It is well known that very special $\Gamma$–spaces and grouplike ${E}_{\infty }$–spaces both model connective spectra. Both these models have equivariant analogues in the case when the group acting is finite. Shimakawa defined the category of equivariant $\Gamma$–spaces and showed that special equivariant $\Gamma$–spaces determine positive equivariant spectra. Costenoble and Waner [Trans. Amer. Math. Soc. 326 (1991) 485-505] showed that grouplike equivariant ${E}_{\infty }$–spaces determine connective equivariant spectra.

We show that with suitable model category structures the category of equivariant $\Gamma$–spaces is Quillen equivalent to the category of equivariant ${E}_{\infty }$–spaces. We define the units of equivariant ring spectra in terms of equivariant $\Gamma$–spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.

Keywords
equivariant infinite loop space, equivariant $\Gamma$–space, equivariant spectra
Mathematical Subject Classification 2000
Primary: 55P91, 55P42
Secondary: 55P47, 55P48