The Picard group of the category of $C_n$-equivariant stable homotopy theory

TL;DR

This paper investigates the Picard group of $G$-spectra in equivariant stable homotopy theory, proving surjectivity of a key map for cyclic groups and describing the group explicitly, with implications for homology and cohomology calculations.

Contribution

It proves the surjectivity of the map from the real representation ring to the Picard group for cyclic groups and explicitly describes the Picard group in that case.

Findings

The map $RO(G) o { m Pic}(Sp^G)$ is surjective for cyclic groups.

The Picard group ${ m Pic}(Sp^G)$ is explicitly described for cyclic groups.

Homology and cohomology with Mackey functor coefficients do not detect certain parts of ${ m Pic}(Sp^G)$.

Abstract

For a finite group $G$, there is a map $RO(G) \to {\rm Pic}(Sp^G)$ from the real representation ring of $G$ to the Picard group of $G$-spectra. This map is not known to be surjective in general, but we prove that when $G$ is cyclic this map is indeed surjective and in that case we describe ${\rm Pic}(Sp^G)$ explicitly. We also show that for an arbitrary finite group $G$ homology and cohomology with coefficients in a cohomological Mackey functor do not see the part of ${\rm Pic}(Sp^G)$ coming from the Picard group of the Burnside ring. Hence these homology and cohomology calculations can be graded on a smaller group.