The $C_2$-equivariant ordinary cohomology of complex quadrics II: The symmetric case

TL;DR

This paper computes the $C_2$-equivariant cohomology of smooth symmetric complex quadrics, revealing new structural properties and examples where cohomology includes summands from the free orbit, advancing understanding in equivariant topology.

Contribution

It provides the first detailed calculation of the equivariant cohomology of symmetric complex quadrics with novel summand structures, expanding the class of known examples.

Findings

Cohomology includes summands from the free orbit $C_2/e$

First example where cohomology is not just shifted copies of a point

Reveals new structural properties of equivariant cohomology

Abstract

In this, the second of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth symmetric quadrics graded on the representation ring of $\Pi BU(1)$ and with coefficients in the Burnside Mackey functor. These calculations exhibit various interesting properties, including the first naturally occurring example we are aware of where the cohomology is not just the sum of shifted copies of the cohomology of a point, but also has summands that are shifted copies of the cohomology of the free orbit $C_2/e$.