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Abstract: In this paper we study cobordism categories consisting of manifolds which areendowed with geometric structure. Examples of such geometric structures includesymplectic structures, flat connections on principal bundles, and complexstructures along with a holomorphic map to a target complex manifold. A generalnotion of -geometric structure- is defined using sheaf theoretic constructions.Our main theorem is the identification of the homotopy type of such cobordismcategories in terms of certain Thom spectra. This extends work ofGalatius-Madsen-Tillmann-Weiss who identify the homotopy type of cobordismcategories of manifolds with fiberwise structures on their tangent bundles.Interpretations of the main theorem are discussed which have relevance totopological field theories, moduli spaces of geometric structures, andh-principles. Applications of the main theorem to various examples of interestin geometry, particularly holomorphic curves, are elaborated upon.

Autor: David Ayala


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