Diffeomorphisms and the topology of positive scalar curvature

几何拓扑学的一个基本目标是了解一个基础拓扑和它所支持的几何类型之间的相互作用。对于这个项目，“基础拓扑”是由一个光滑流形$M$给出的，并且这样一个流形的微分同胚群也是这个“基础拓扑”的一部分。我们关于微分同胚群的许多知识都使用同伦理论。我们项目的一部分将涉及这个同伦的显式几何构造，从理论上给出信息。然后，我们的目标是将其用于新的(次要)构造和应用。我们计划集中研究的应用类型是研究微分同胚群在正数量曲率的度量空间pos(M)在M上的作用。这样的应用提出了以下研究挑战和问题：*获得关于diff(M)的适当信息。经典地说，这涉及到代数和几何拓扑学中的不稳定同伦理论、光滑化理论和其他工具。*找到工具来理解作用对pos(M)的影响，并将它们应用于手头的情况。这将涉及狄拉克算符的指数理论和正标量曲率度量的Gromov-Lawson手术技术。*为pos(M)开发协调空间变量，它允许更灵活的结构和更适合计算，并将它们与真实的pos(M)进行比较。*找到显式的几何结构。我们这样做是为了更好地从理论上理解关于pos(M)的抽象同伦理论信息的几何内容，并希望得到新的附加工具。*发展关于diff(M)在pos(M)上作用的刚性结果。我们预计，这一行动将通过Cobordism类别进行考虑。对从diff(M)到这个余界范畴的通道的核的计算应该表明diff(M)的同伦群的大子群平凡地作用于pos(M)的同伦群。*最后，我们将讨论非紧流形M上(紧支撑的)微分同态的作用在多大程度上引起pos(M)中的非平凡类的问题，其中pos(M)现在表示完备的一致正标量曲度量的空间。这将涉及到“推向无穷大”所带来的灵活性和粗略指数理论的新工具之间的相互作用。

One of the fundamental goals of geometric topology is to understand the interplay between an underlying topology and the types of geometry supported by this topology.For this project, the “underlying topology” is given by a smooth manifold $M$, and also the diffeomorphism group of such a manifold is part of this “underlying topology”. A lot of our knowledge about the diffeomorphism group uses homotopy theory. One part of our project will be concerned with explicit geometric constructions of this homotopy theoretically given information. We then aim at exploiting this for new (secondary) constructions and applications. The type of applications we plan to concentrate on is the study the action of the diffeomorphism group on the space of metrics of positive scalar curvature Pos(M) on M. Such applications raise the following research challenges and questions:* get appropriate information about Diff(M). Classically, this involves unstable homotopy theory, smoothing theory and other tools from algebraic and geometric topology.* find tools to understand the effect of the action on Pos(M), and apply them to the situation at hand. This will involve index theory of Dirac operators and the Gromov-Lawson surgery technique for metrics of positive scalar curvature.* develop concordance space variants for Pos(M) which allow from more flexible constructions and are more appropriate for calculations, and compare them to the true Pos(M).*find explicit geometric constructions. We do this to get a better theoretic understanding of the geometric content of the abstract homotopy theoretic information about Pos(M), and also with the hope to arrive at new and additional tools.* develop rigidity results on the action of Diff(M) on Pos(M). We expect that this action factors through a cobordism category. Computation of the kernel of the passage from Diff(M) to this cobordism category should show that large subgroups of the homotopy groups of Diff(M) act trivially on the homotopy groups of Pos(M).*Finally, we will address the question to which extent the action of (compactly supported) diffeomorphisms on a non-compact manifold M gives rise to non-trivial classes in Pos(M), where Pos(M) now denotes the space of complete uniformly positively scalar curved metrics. This will involve an interplay between the flexibility given by “pushing off to infinity”' and new tools from coarse index theory.