Diffeomorphisms and the topology of positive scalar curvature

几何拓扑的基本目标之一是理解底层拓扑与该拓扑所支持的几何类型之间的相互作用。对于这个方案，“底层拓扑”由光滑流形$M$给出，并且这种流形的微分同构群是这个“底层拓扑”的一部分。我们对微分同构群的很多认识都是利用同伦理论。我们计划的一部分将会关注这个同伦的显式几何结构在给定信息的理论。然后我们的目标是将其用于新的（二级）结构和应用。我们计划集中研究M上的微分同构群对正标量曲率Pos(M)的度量空间的作用。这种应用提出了以下的研究挑战和问题：*获得关于Diff(M)的适当信息。经典地，这涉及到不稳定同伦理论，平滑理论和其他工具从代数和几何拓扑。*寻找工具来了解行动对Pos(M)的影响，并将其应用于手头的情况。这将涉及狄拉克算子的指标理论和正标量曲率度量的Gromov-Lawson手术技术。*为词序(M)开发更灵活的结构和更适合计算的一致性空间变体，并将其与真正的词序(M)进行比较。*找到明确的几何结构。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，并希望得到新的和额外的工具。*开发Diff(M)对Pos(M)作用的刚度结果。我们期望这一行动通过协同范畴发挥作用。从Diff(M)到这个协范畴的通道核的计算应该表明Diff(M)的同伦群的大子群对Pos(M)的同伦群起着微不足道的作用。*最后，我们将讨论（紧支持的）微分同态在非紧流形M上的作用在多大程度上产生Pos(M)中的非平凡类，其中Pos(M)现在表示完全一致正标量弯曲度量空间。这将涉及到“推到无穷大”所带来的灵活性和来自粗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.