2024/06/01 by Markus Hausmann, Hausmann, Markus, Stefan Schwede +1
Decision Sciences · Mathematics · #55N22 #55N91 #55P91 #57R85 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Fuzzy and Soft Set Theory #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2406.00404
openalex publication_date 2024/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 el2RO-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 el2RO-algebras, so examples abound. Inverting the inverse Thom classes yields a global 2-torsion group law. In this sense, our oriented el2RO-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 el2RO-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 el2RO-algebra with an invertible orientation; that Bredon homology with constant mod 2 coefficients is an initial el2RO-algebra with an additive orientation; and that Borel equivariant homology with mod 2 coefficients is an initial el2RO-algebra with an orientation that is both additive and invertible.