FRG: Collaborative Research: Floer homotopy theory

Topology is the study of those properties of shapes that are unchanged by stretching and bending. Over the years, mathematicians have developed various topological invariants, or in other words, quantities that are associated to shapes and can distinguish between those that have different properties. Homology is a well-known such invariant, which can be associated to any multi-dimensional shape, and which is a quantitative measure of the number of holes in a space. A circle, for instance, has only a one-dimensional hole, whereas the surface of a doughnut has two one-dimensional holes, a meridian and a longitude, that are not filled in by the surface itself, and an additional two-dimensional hole. Floer homology is a more refined invariant that is responsible for some of the most important recent advances in the study of knotted closed loops in space, three-dimensional shapes, and shapes with a geometry known as a symplectic structure that is exhibited by phase spaces in classical mechanics. This project brings together several researchers working in different areas of topology and geometry to study Floer homology. The main goal of the project is the following: To every knot, three-dimensional shape, or symplectic shape, one should associate a different object, called a Floer space or a Floer homotopy type, whose (ordinary) homology is the Floer homology of the initial shape. This has been accomplished so far in a limited number of cases. A general theory of Floer spaces will lead to new advances in several areas. Furthermore, the study of Floer spaces will be based on techniques from a subfield of topology called homotopy theory. This project will create a community of scholars at the interface of these current and extremely research active areas of mathematics.

Floer homology is a fundamental tool in geometry and topology, whose applications range from the Arnold conjecture to the surgery characterization of various knots. Floer homology has also laid the basis for completely unexpected interconnections between algebraic and symplectic geometry in the form of homological mirror symmetry. Floer homotopy theory, an extension to spaces rather than homology groups, has been implemented in a small number of cases, leading to significant applications, for example, the resolution of the triangulation conjecture in high dimensions and work on immersed Lagrangian spheres. Further, the ideas behind Floer homotopy inspired the construction of a Khovanov homotopy type associated to knots in the three-sphere. The main scientific goal of this project is to give a general construction of Floer homotopy. The necessary foundational work will build upon recent advances in multiple areas. These include the conceptual advances in equivariant stable homotopy theory stemming from the resolution of Kervaire invariant one problem, and the development of new approaches to define virtual fundamental classes in Floer theory. The project aims to put the homotopical and homological variants of Floer theory on equal footing. As a consequence, new applications in both symplectic and low-dimensional topology are anticipated, for example: (i) a spectral Fukaya category associated to a symplectic manifold will be constructed; (ii) the Heegaard Floer theory of Ozsvath and Szabo will be used to produce a computable invariant parallel to the celebrated Bauer-Furuta invariant for four-manifolds; (iii) Seiberg-Witten Floer homotopy types will be studied using the tools of equivariant stable homotopy theory; and (iv) the Khovanov homotopy type will be extended to give invariants of knot cobordisms and tangles.