Group Actions and Floer-Theoretic Invariants

这是一个纯数学项目，专注于低维和辛拓扑。它有两个广泛的目标：研究三维同调配边群，与三维球面具有许多代数特征的空间集合，以及理解流形之间的面积保持映射。从哲学上讲，这些目标是统一的愿望，以了解灵活性或刚性的几何性质的空间。理解同调配边群是拓扑学中的一个主要动机问题;这些群在三维中的复杂性问题与高维拓扑中的深层结构问题有关。类似地，保面积（辛）映射的研究是辛几何的中心问题之一，并且与物理学有着非平凡的联系。PI还将研究与本研究相关的结一致性不变量的应用。由于这个项目的动机之一是，所使用的工具是特别计算访问，PI将积极寻求参与本科生和其他年轻的研究人员在这项工作。这个项目的工具是Floer理论不变量的等变版本。 有两个主要方案。第一个涉及最近构造的对合版本的三流形不变的Heegaard Floer同源，它给出了两个新的同源配边和结协调不变量。利用这个不变量，PI将研究同调配边群和结的美容手术。对合Heegaard Floer同源性是朝着构建Pin（2）-等变Heegaard Floer同源性的长期目标迈出的第一步，PI将继续努力。这是可取的，鉴于最近的进展，使用Pin（2）-等变版本的Seiberg-Witten Floer同源性，一个类似的理论。对于第二个，PI建议构造一个Serre谱序列的拉格朗日Floer上同调的某些辛纤维。从这个谱序列中产生的关系被期望给出关于辛映射类群的信息;此外，由于辛和低维拓扑中的许多不变量可以用拉格朗日弗洛尔上同调来表示，谱序列本身具有潜在的广泛后果。其他目标包括理解一般李群的等变拉格朗日弗洛尔上同调，它也有许多理论应用;例如，Z_2的现有理论的Z_p版本将意味着判定具有稳定平凡化切丛的流形的辛同胚具有无限阶的标准。

This is a project in pure mathematics focusing on low-dimensional and symplectic topology. It has two broadly constructed goals: to study the three-dimensional homology cobordism groups, sets of spaces with many algebraic features in common with the three-dimensional sphere, and to understand area-preserving maps between manifolds. Philosophically speaking, these goals are united by a desire to understand the flexibility or rigidity of geometric qualities of spaces. Understanding the homology cobordism groups is a major motivating question in topology; questions of the complexity of these groups in three dimensions are tied to deep structural issues in higher-dimensional topology. Similarly, the study of area-preserving (symplectic) maps is one of the central issues of symplectic geometry, and has nontrivial connections to physics. The PI will also study applications of invariants of knot concordance which arise in connection with this research. Since one motivation for this project is that the tools used are particularly computationally accessible, the PI will actively seek to involve undergraduate and other young researchers in this work. The tools of this project are equivariant versions of Floer-theoretic invariants. There are two main programs. The first involves a recently-constructed involutive version of the three-manifold invariant Heegaard Floer homology, which gives two new homology cobordism and knot concordance invariants. Using this invariant, the PI will study the homology cobordism group and cosmetic surgeries on knots. Involutive Heegaard Floer homology is a first step toward the long-term goal of constructing Pin(2)-equivariant Heegaard Floer homology, which the PI will continue to work toward. This is desirable in light of recent progress made using Pin(2)-equivariant version of Seiberg-Witten Floer homology, an analogous theory. For the second, the PI proposes to construct a Serre spectral sequence for the Lagrangian Floer cohomology of certain symplectic fibrations. Relationships arising from this spectral sequence are expected to give information about the symplectic mapping class group; furthermore, since many invariants in symplectic and low-dimensional topology can be formulated in terms of Lagrangian Floer cohomology, the spectral sequence itself has potentially broad consequences. Other goals include understanding equivariant Lagrangian Floer cohomology for general Lie groups, which also has many theoretical applications; a Z_p version of existing theory for Z_2, for example, would imply a criterion for deciding that a symplectomorphism of a manifold with stably trivialized tangent bundle has infinite order.