Some thoughts about monopole h-invariants

Date: 2022-02-24

Time: 14:15 - 16:00


Stefan Behrens


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.