$C_{2^n}$-equivariant rational stable stems and characteristic classes

TL;DR

This paper computes rational $C_{2^n}$-equivariant stable stems, provides minimal presentations for certain equivariant cohomology groups, and analyzes the conditions under which maximal torus inclusions induce isomorphisms in these cohomology groups.

Contribution

It introduces explicit calculations of rational $C_{2^n}$-equivariant stable stems and cohomology of classifying spaces, and characterizes when maximal torus inclusions induce isomorphisms.

Findings

Computed rational $C_{2^n}$-equivariant stable stems.

Provided minimal presentations for $RO(C_{2^n})$-graded Bredon cohomology.

Identified conditions for maximal torus inclusions to induce isomorphisms.

Abstract

In this short note, we compute the rational $C_{2^n}$-equivariant stable stems and give minimal presentations for the $RO(C_{2^n})$-graded Bredon cohomology of the equivariant classifying spaces $B_{C_{2^n}}S^1$ and $B_{C_{2^n}}\Sigma_2$ over the rational Burnside functor $A_{\mathbf Q}$. We also examine for which compact Lie groups $L$ the maximal torus inclusion $T\to L$ induces an isomorphism from $H^*_{C_{2^n}}(B_{C_{2^n}}L;A_{\mathbf Q})$ onto the fixed points of $H^*_{C_{2^n}}(B_{C_{2^n}}T;A_{\mathbf Q})$ under the Weyl group action. We prove that this holds for $L=U(m)$ and any $n,m\ge 1$ but does not hold for $L=SU(2)$ and $n>1$.