Equivariant Anderson duality and Mackey functor duality

TL;DR

This paper demonstrates that certain $bZ/2$-equivariant Morava K-theories with reality are self-dual under equivariant Anderson duality, revealing symmetries in equivariant cohomology via Mackey functor duality.

Contribution

It establishes the self-duality of $bZ/2$-equivariant Morava K-theories with reality and interprets symmetries in equivariant cohomology through Mackey functor duality.

Findings

Self-duality of Morava K-theories with reality

Universal coefficients exact sequence in equivariant K-theory

Interpretation of cohomology symmetries via Mackey functor duality

Abstract

We show that the $\mathbb{Z}/2$-equivariant Morava K-theories with reality (as defined by Hu) are self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in Morava K-theory with reality. As a particular example, we recover the self-duality of the spectrum $KO$. The study of $\mathbb{Z}/2$-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of $RO(\mathbb{Z}/2)$-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.