Spin obstructions to metrics of positive scalar curvature on nonspin manifolds

在光滑流形上构造阻碍正标量曲率度量存在的障碍，是近几十年来微分几何中的一个重要课题。在闭合自旋流形上，这种度量存在的最大障碍是基于自旋狄拉克算子的指标理论。这种方法的主要限制是它只适用于自旋流形，或者至少适用于具有自旋盖的流形。这个项目的主要目标是，在某些情况下，将狄拉克算子方法扩展到流形没有自旋结构的情况。特别地，我们考虑以下情况：M是一个封闭的面向黎曼流形，V\子集M是一个封闭子集，表示第二类Stiefel-Whitney类的Poincar\ e对偶。一个拓扑不变量存储在远离V的地方，更准确地说，是两个束E_0， E_1，它们具有同构的典型纤维，被支持在边界为L\子集m的流形内部。L的双D是一个封闭的自旋流形，具有一个束E，通过在D的一半上附加E_1，在另一半上附加E_0而得到。我们的拓扑不变量是D上与e束扭曲的自旋狄拉克算子的指标。这个项目的主要问题是，至少在适当的条件下，这个算子的指标是否阻碍m上正标量曲率的度量的存在。我们考虑的主要几何情况是闭合流形M_1和M_2的连通和，其中M_1存储着指标障碍。当$V$是一个封闭的余维二子流形时，我们计划发展一个指标理论，使我们能够将存储在集合L中的信息编码为m中V的补上的算子。这与Gromov和Lawson在非紧流形上的工作类似。对于V附近算子的分析，我们计划使用Degeratu和Stern的结果，以便至少在合适的条件下，得到一个消失定理。我们计划发展的用于研究非自旋流形的指标理论也适用于自旋流形。特别地，它涉及到Gromov提出的两个问题，即在存在到球体的合适映射的条件下，正标量曲率的度量是否存在。本项目的另一个目标是研究非自旋环境下的可面积放大度量。从Gromov和Lawson的工作中我们知道，在自旋流形上不存在标量曲率一致为正的完全的可面积放大度量。为了将这一结果推广到非自旋情况，我计划研究的方法是基于狄拉克方法和最小超曲面方法之间的相互作用。

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.