Date & Time: Monday, January 11, 2010, 17:00-18:00
Venue: Ramanujan Hall
Title: Equivariant Infinite Loop Spaces
Speaker: Rekha Santhanam, Johns Hopkins University
Abstract: Generalized equivariant cohomology theories such as equivariant K-theory and equivariant singular cohomology are represented by equivariant spectra. \textit{Connective} equivariant spectra have several space level models in the case when the group acting is a finite.
We describe two such models, equivariant $\Gamma$-spaces defined by Shimakawa and equivariant $E_\infty$-spaces defined by Constenoble, May and Waner. We show that they are Quillen equivalent and therefore, both the categories have the same homotopy theory.
In the non-equivariant case, the twistings of a cohomology theory, for example Twisted K-theory (which finds applications in String theory) are classified by the units of the ring spectrum representing the cohomology theory. Motivated by the notion of twisted equivariant K-theory, we define the group of \textit{units } of equivariant ring spectra in terms of equivariant $\Gamma$-spaces.
We then give a generalization of Segal's construction which gives a method to write down examples of equivariant $\Gamma$-spaces starting with symmetric monoidal categories enriched with the group action.