Group Actions and Floer-Theoretic Invariants

Michigan State University, East Lansing MI

Investigators

Abstract

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.