The universal property of bordism of commuting involutions

TL;DR

This paper introduces a universal algebraic framework called oriented el$_2^{RO}$-algebras to describe the structure of equivariant bordism rings with commuting involutions, unifying various equivariant homology theories.

Contribution

It defines oriented el$_2^{RO}$-algebras and proves that equivariant bordism and several homology theories are initial objects in this category, revealing their universal properties.

Findings

Equivariant bordism for elementary abelian 2-groups is an initial oriented el$_2^{RO}$-algebra.

Stable equivariant bordism is an initial el$_2^{RO}$-algebra with an invertible orientation.

Bredon and Borel homology with mod 2 coefficients are initial el$_2^{RO}$-algebras with specific orientations.

Abstract

We propose a formalism to capture the structure of the equivariant bordism rings of smooth manifolds with commuting involutions. We introduce the concept of an oriented el$_2^{RO}$-algebra, an algebraic structure featuring representation graded rings for all elementary abelian 2-groups, connected by restriction homomorphisms, a pre-Euler class, and an inverse Thom class; this data is subject to one exactness property. Besides equivariant bordism, oriented global ring spectra also give rise to oriented el$_2^{RO}$-algebras, so examples abound. Inverting the inverse Thom classes yields a global 2-torsion group law. In this sense, our oriented el$_2^{RO}$-algebras are delocalized generalizations of global 2-torsion group laws. Our main result shows that equivariant bordism for elementary abelian 2-groups is an initial oriented el$_2^{RO}$-algebra. Several other interesting equivariant homology theories can also be characterized, on elementary abelian 2-groups, by similar universal properties. We prove that stable equivariant bordism is an initial el$_2^{RO}$-algebra with an invertible orientation; that Bredon homology with constant mod 2 coefficients is an initial el$_2^{RO}$-algebra with an additive orientation; and that Borel equivariant homology with mod 2 coefficients is an initial el$_2^{RO}$-algebra with an orientation that is both additive and invertible.