Cobordism Category of Manifolds With Baas-Sullivan Singularities, Part I

TL;DR

This paper constructs a cobordism category for manifolds with Baas-Sullivan singularities and identifies its classifying space's homotopy type with an infinite loop space related to a specific spectrum, extending classical bordism theories.

Contribution

It introduces a cobordism category for manifolds with Baas-Sullivan singularities and relates its classifying space to a spectrum, generalizing classical bordism results.

Findings

Homotopy type of classifying space identified with an infinite loop-space.

Established an analogue of the Bockstein-Sullivan exact couple.

Extended classical bordism theories to manifolds with singularities.

Abstract

For a fixed closed manifold $P$, we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type $P$. Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectrum $\text{MT}(d)$ introduced in [arXiv:math/0605249]. We obtain an analogue of the Bockstein-Sullivan exact couple that arises between the classical bordism theories $MO$ and $MO_{P}$ on the level of cobordism categories.