In this paper we study cobordism categories consisting of manifolds which are
endowed with geometric structure. Examples of such geometric structures include
symplectic structures, flat connections on principal bundles, and complex
structures along with a holomorphic map to a target complex manifold. A general
notion of "geometric structure" is defined using sheaf theoretic constructions.
Our main theorem is the identification of the homotopy type of such cobordism
categories in terms of certain Thom spectra. This extends work of
Galatius-Madsen-Tillmann-Weiss who identify the homotopy type of cobordism
categories of manifolds with fiberwise structures on their tangent bundles.
Interpretations of the main theorem are discussed which have relevance to
topological field theories, moduli spaces of geometric structures, and
h-principles. Applications of the main theorem to various examples of interest
in geometry, particularly holomorphic curves, are elaborated upon.Comment: 82 pages. Second version with remarks on higher category approaches
and various minor correction