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Spin obstructions to metrics of positive scalar curvature on nonspin manifolds

Suppose $$M$$ is a closed oriented Riemannian manifold and V is a closed subset of M representing the Poincaré dual of the second Stiefel-Whitney class. We also suppose to have a topological invariant stored away from $$V$$More precisely, we suppose there are two bundles $$E_0$$, $$E_1$$ with isomorphic typical fibers that are supported in the interior of a manifold with boundary $$L\subset M$$Observe that the double $$L_D=L\cup_{\partial L}L^-$$ of $$L$$is a closed spin manifold endowed with a bundle $$E$$coinciding with $$E_1$$ on $$L$$and with and with $$E_0$$ on $$L^-$$Our topological invariant is the index $$\text{ind}( D_{L_D,E})$$ of the spin Dirac operator on $$L_D$$ twisted with the bundle $$E$$.

The main goal of this project is to give conditions such that the invariant $$\text{ind}( D_{L_D,E})$$ is an obstruction to the existence of metrics of positive scalar curvature on $$M$$. The analytic tools we plan to use are based on the analysis of the spin Dirac operator on the incomplete manifold $$M\setminus V$$ , using a potential and a rescaling function to control the behavior of the Dirac operator near the deleted subset $$V$$. The main geometric situation we have in mind is the connected sum $$M_1\# M_2$$, with $$M_1$$ a closed spin manifold storing the Dirac obstruction.

When $$V$$ is a closed codimension two submanifold, the analytic machinery has already been developed by the author, making use of the distance function to the submanifold $$V$$ to construct the rescaling function and the potential. These analytic tools also have applications to the study of metrics of positive scalar curvature on a compact manifolds with boundary $$X$$. More precisely, it allows to find an upper bound for the distance between the region where the topological information is stored and the boundary of the manifold, knowing a positive lower bound for $$\text{scal}(X)$$. This answers questions recently asked by Gromov. Initially, it was the second main goal of this project.

## Team Members

Dr. Simone Cecchini