On the $KU_G$-local equivariant sphere

TL;DR

This paper computes the zeroth homotopy Green functor and Tambara functor structures of localized equivariant sphere spectra with respect to equivariant complex K-theory, focusing on odd p-groups.

Contribution

It provides explicit calculations of the zeroth homotopy Green and Tambara functors for localized equivariant spheres in the case of odd p-groups, advancing understanding of equivariant stable homotopy theory.

Findings

Calculated the zeroth homotopy Green functor for odd p-groups.

Determined the zeroth homotopy Tambara functor for odd cyclic p-groups.

Enhanced the understanding of equivariant sphere spectra localization.

Abstract

Equivariant complex $K$-theory and the equivariant sphere spectrum are two of the most fundamental equivariant spectra. For an odd $p$-group, we calculate the zeroth homotopy Green functor of the localization of the equivariant sphere spectrum with respect to equivariant complex $K$-theory. Further, we calculate the zeroth homotopy Tambara functor structure in the case of odd cyclic $p$-groups.