Gauge-Theoretic Floer Homologies

Organized by Gary and Reyna.

This is a learning seminar with the goal of eventually understanding Manolescu's recent work in Floer homology. In particular, we discuss gauge-theoretic methods in low-dimensional topology to classical problems concerning high-dimensional manifolds. The central focus is on Seiberg-Witten Floer homology, a powerful invariant for 3-manifolds originally developed by Kronheimer and Mrowka. Ciprian Manolescu famously resolved the long-standing Triangulation Conjecture by developing a Pin(2)-equivariant version of this theory. This refined equivariant structure allowed for the definition of a novel integer-valued invariant that obstructs triangulability, proving the existence of non-triangulable manifolds in dimensions five and higher. The analytic foundations of this work can be further understood through the modern formalism of the Seiberg-Witten-Floer stable homotopy type, which leverages finite-dimensional approximation schemes and the Conley index to construct the invariants.

References

A more comprehensive reading list for this seminar will appear soon. For now, our main resource will be:

• Kronheimer & Mrowka's Monopoles and Three-Manifolds

Later, we will dive into a bunch of papers (TBD).

Schedule

We meet in Wachenheim on Fridays from 7-9 and Sundays time TBD.