The Equivariant Cobordism Category

TL;DR

This paper introduces an equivariant cobordism category for finite groups, characterizing its classifying space's homotopy type as fixed points of an infinite loop space of an equivariant spectrum, advancing the understanding of equivariant topology.

Contribution

It defines the equivariant cobordism category for finite groups and identifies its classifying space's homotopy type in terms of equivariant spectra.

Findings

Homotopy type of the classifying space is identified as fixed points of an equivariant spectrum's infinite loop space.

Provides a new categorical framework for studying equivariant cobordisms.

Connects geometric objects with algebraic topological invariants in equivariant settings.

Abstract

For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (i.e. geometric realization of its simplicial nerve) as the fixed points of the infinite loop space of an equivariant spectrum.