Group Actions and Floer-Theoretic Invariants

这是一个纯数学的项目，专注于低维和辛拓扑。它有两个广泛构建的目标：研究三维同调余边群，即具有许多与三维球面相同的代数特征的空间集，以及理解流形之间的保面积映射。从哲学上讲，这些目标是由理解空间几何性质的灵活性或刚性的愿望统一起来的。理解同调余边群是拓扑学中的一个主要激励问题；这些群在三维空间中的复杂性问题与更高维拓扑中的深层结构问题有关。同样，保面积(辛)映射的研究是辛几何的中心问题之一，与物理学有很大的联系。PI还将研究与这项研究相关的纽结一致性不变量的应用。由于这个项目的一个动机是所使用的工具特别容易通过计算获得，所以PI将积极寻求让本科生和其他年轻的研究人员参与这项工作。这个项目的工具是Floer理论不变量的等变版本。有两个主要的计划。第一个涉及最近构造的三维流形不变量Heegaard Floer同调的对合形式，它给出了两个新的同调协边和纽结协调不变量。利用这个不变量，PI将研究纽结上的同调余边群和整形外科手术。对合Heegaard Floer同调是朝着构建Pin(2)等变Heegaard Floer同调的长期目标迈出的第一步，PI将继续朝着这一目标努力。鉴于最近使用类似理论Seiberg-Witten Floer同调的Pin(2)等变版本所取得的进展，这是可取的。对于第二种情形，PI建议为某些辛纤维的拉格朗日Floer上同调构造一个Serre谱序列。由此谱序列产生的关系有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Floer上同调来表示，因此谱序列本身具有潜在的广泛结果。其他目标包括理解一般李群的等变Lagrangian Floer上同调，这也有许多理论应用；例如，Z2的现有理论的Zp版本将意味着一个判定具有稳定平凡切丛的流形的辛同态具有无穷阶的准则。

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.