Derived Geometry and the Integrability Problem for G-Structures

Abstract

In this thesis, we study the integrability problem for G-structures. Broadly speaking, this is the problem of determining topological obstructions to the existence of principal G-subbundles of the frame bundle of a manifold, subject to certain differential equations. We begin this investigation by introducing general methods from homological algebra used to obtain cohomological obstructions to the existence of solutions to certain geometric problems. This leads us to a precise analogy between deformation theory and the formal integrability properties of partial differential equations. Along the way, we prove the following differential-geometric analogue of a well-known result from derived algebraic geometry. Namely, the cohomology of the normalized complex associated to the simplicial object obtained by tensoring the simplicial set correpsonding to the circle with the smooth algebra of smooth functions on a smooth manifold is precisely the algebra of differential forms.

We also identify the infinitesimal generator of the natural circle-action with the de Rham differential. As a short corollary we obtain a natural isomorphism between the dual of this algebra and the algebra of poly-vector-fields, leading to a comparison between the Gerstenhaber bracket of Hochshild cohomology and the Schouten bracket. These two results are well-known in derived algebraic geometry and are folk-lore in differential geometry, where we were unable to find an explicit proof in the literature. In the end, this machinery is used to provide what the author believes is a new perspective on the integrability problem for G-structures.