Genuine Equivariant Floer Homotopy Theory and Topological Hochschild Homology

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Semon Rezchikov, Princeton University

Online Talk

Work initiated by Cohen-Jones-Segal (CJS) assigns stable homotopy types to (appropriately decorated) flow categories, which axiomatize the behavior of gradient flows of Morse functions and can be extracted from Floer-theoretic PDEs. I will explain how to refine this construction to assign genuine equivariant stable homotopy types to equivariant flow categories. Unfortunately, equivariant Morse functions cannot be perturbed to nearby equivariant Morse-Smale functions, so this work requires a serious rethinking of the CJS construction to allow the Morse function to no longer be Morse-Smale. This reformulation interacts nicely with equivariant stable homotopy theory, especially with the geometric fixed points functor, and allows one to construct a (genuine) cyclotomic spectrum in the sense of Hesselholt-Madsen and Nikolaus-Scholze which refines symplectic cohomology, a well known invariant of symplectic manifolds with boundary such as affine algebraic varieties and cotangent bundles. The construction suggests systematic arithmetic refinements of symplectic invariants, as well as noncommutative analogs of results in arithmetic geometry. In particular, a `picture proof' suggests a generalization of a well-known formula of Cartier to the setting of noncommutative algebras, which turns out to have an enumerative meaning when specialized to the Fukaya category of a symplectic manifold.