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
    Researcher, Project leader
    Georg-August Universität Göttingen

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