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Stefan Behrens: Some thoughts about monopole h-invariants

Time: Thu 2022-02-24 14.15 - 16.00

Location: Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

Video link: Meeting ID: 921 756 1880

Participating: Stefan Behrens (Bielefeld University)

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Abstract: The monopole h-invariants are numerical invariants of closed, oriented 3-manifolds with the same rational homology as the 3-sphere. They were first defined by Froyshov in his work on Seiberg-Witten theory on 3-manifolds and shown to give restrictions on the possible intersection forms of bounding 4-manifolds. Nowadays, the h-invariants are commonly extracted from the monopole Floer homology package constructed by Kronheimer and Mrowka. It is a long standing question whether or not the h-invariants depend on the choice of coefficient ring. The goal of this talk is to discuss this problem using Manolescu's homotopy theoretic approach to 3d Seiberg–Witten theory, which recovers monopole Floer homology from an \(S^1\)-equivariant stable homotopy type. These Seiberg–Witten–Floer homotopy types are known to have a few special properties and the definition of h-invariants extends to abstract homotopy types with these properties. I will discuss examples of such homotopy types whose h-invariants exhibit non-trivial coefficient dependence. However, it remains an open question whether or not these homotopy types are realized by 3-manifolds.