An equivariant Quillen theorem

TL;DR

This paper extends Quillen's classical theorem to the equivariant setting, identifying the $bZ/2$-equivariant unitary bordism ring with the $bZ/2$-equivariant Lazard ring, combining homotopy computations and algebraic investigations.

Contribution

It provides the first equivariant generalization of Quillen's theorem by explicitly identifying the equivariant bordism and Lazard rings.

Findings

Identified the $bZ/2$-equivariant unitary bordism ring with the $bZ/2$-equivariant Lazard ring.

Computed the homotopy theoretic $bZ/2$-equivariant unitary bordism ring.

Analyzed the structure of the $bZ/2$-equivariant Lazard ring.

Abstract

A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring, introduced by tom Dieck (1970), with the $\mathbb{Z}/2$-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). Our proof combines a computation of the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring due to Strickland (2001) with a detailed investigation of the $\mathbb{Z}/2$-equivariant Lazard ring.