Spin obstructions to metrics of positive scalar curvature on nonspin manifolds

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It has been an important topic in differential geometry in recent decades to construct obstructions to the existence of metrics of positive scalar curvature on a smooth manifold. On closed spin manifolds, the most powerful obstructions to the existence of such metrics areIt based on the index theory for the spin-Dirac operator. The main restriction of this method is that it applies only to spin manifolds, or at least to manifolds with a spin cover. The main goal of this project is to extend, in some cases, the Dirac operator method to the case when the manifold does not have a spin structure. In particular, we consider the following situation: M is a closed oriented Riemannian manifold and V\subset M is a closed subset representing the Poincar\'e dual of the second Stiefel-Whitney class. A topological invariant is stored away from $V$, more precisely two bundles E_0, E_1 with isomorphic typical fibers that are supported in the interior of a manifold with boundary L\subset M. The double D of L is a closed spin manifold endowed with a bundle E obtained by attaching E_1 on one half of D and E_0 on the other half. Our topological invariant is the index of the spin Dirac operator on D twisted with the bundle E. The main question o this project is whether, at least under suitable conditions, the index of this operator is an obstruction to the existence of metrics of positive scalar curvature on M. The main geometric situation we have in mind is the connected sum of closed manifolds M_1 and M_2, with M_1 storing the index obstruction.When $V$ is a closed codimension two submanifold, we plan to develop an index theory that allows us to encode the information stored in the set L into an operator on complement of V in M. This is done in analogy with the work of Gromov and Lawson on noncompact manifolds. For the analysis of the operator near V, we plan to use results by Degeratu and Stern in order to obtain, at least under suitable conditions, a vanishing theorem.The index theory we plan to develop to study nonspin manifolds has also applications to spin manifolds. In particular, it is related to two questions asked by Gromov about the existence of metrics of positive scalar curvature under the condition that there exists a suitable map to the sphere.Another objective of this project is the study of area-enlargeable metrics in the nonspin setting. It is well-known from the work of Gromov and Lawson that on spin manifolds there are no complete area-enlargeable metrics whose scalar curvature is uniformly positive. In order to extend this result to the nonspin case, the approach I plan to investigate is based on the interplay between the Dirac method and the minimal hypersurface method.