Units of equivariant ring spectra

It is well known that very special $Γ$ –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 $Γ$ –spaces and showed that special equivariant $Γ$ –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 $Γ$ –spaces is Quillen equivalent to the category of equivariant $E_{\infty}$ –spaces. We define the units of equivariant ring spectra in terms of equivariant $Γ$ –spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.

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Keywords

equivariant infinite loop space, equivariant $\Gamma$–space, equivariant spectra

Mathematical Subject Classification 2000

Primary: 55P91, 55P42

Secondary: 55P47, 55P48

References

Publication

Received: 27 December 2009

Revised: 1 February 2011

Accepted: 21 February 2011

Published: 17 May 2011

Authors

Rekha Santhanam
Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Kanpur UP 208016 India

Volume 11, issue 3 (2011)

Rekha Santhanam

Abstract