Cobordism categories of manifolds with Baas-Sullivan singularities, Part 2

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For a given list of closed manifolds Σₖ=(P₁,...,Pₖ), we construct a cobordism category CobdΣₖ of embedded manifolds with Baas-Sullivan singularities of type Σₖ. Our main results identify the homotopy type of the classifying spaces BCobdΣₖ of these cobordism categories with that of the infinite loop-space of a certain spectrum, MTΣₖ(d). Our results generalize of the work of Galatius, Madsen, Tillmann, and Weiss from arXiv:math/0605249. It also generalizes the work of Genauer on the cobordism categories of manifolds with corners from arXiv:0810.0581.