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Topology/Geometry Seminar

The bar flavor of monopole Floer homology and homological algebra

Mike Miller Eismeier (Columbia University)

Location:  zoom
Date & time: Tuesday, 21 September 2021 at 11:00AM - 12:00PM

Abstract: Kronheimer and Mrowka define three flavors of monopole Floer homology (HM-to, HM-from, and HM-bar, which are known to be isomorphic to HF^+, HF^-, and HF^oo, respectively). The third of these they show depends on much less than the 3-manifold we plug in: HM-bar(Y) can be computed as the "coupled Morse homology" of the torus T(Y) = T^{b_1 Y}, and show that it only depends on the cohomology ring of Y. They also relate this to what might be called the twisted simplicial homology of said torus T(Y). Over the rationals, Kronheimer and Mrowka show that this is enough to completely determine the homology groups HM-bar(Y). However, their techniques cannot be applied integrally.

I will explain recent joint work of myself and Francesco Lin, where we investigate this twisted simplicial homology for its own sake. By proving a formality property for twisted homology of spaces with torsion-free cohomology, we are able to show that Kronheimer and Mrowka's answer holds over the integers as well (bypassing the spectral sequence and thus avoiding extension problems). Passing through the isomorphism HM-bar = HF^oo, this confirms a conjecture of Oszvath and Szabo. We show that this is lucky to T(Y) and twisted homology may be quite complicated in general.