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

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 $M{O}_P$ on the level of cobordism categories.