Nilpotency in instanton homology, and the framed instanton homology of a surface times a circle

TL;DR

This paper computes the nilpotency degree of a specific endomorphism in instanton Floer homology for surface times circle and determines the framed instanton homology with non-trivial bundle, relating results to moduli spaces of holomorphic bundles.

Contribution

It provides explicit calculations of nilpotency and framed instanton homology for surface times circle, connecting these to the geometry of moduli spaces of holomorphic bundles.

Findings

Nilpotency degree of u^2-64 is computed.

Framed instanton homology with non-trivial bundle is determined.

Results relate to the moduli space of stable rank two holomorphic bundles.

Abstract

In the description of the instanton Floer homology of a surface times a circle due to Mu\~{n}oz, we compute the nilpotency degree of the endomorphism $u^2-64$. We then compute the framed instanton homology of a surface times a circle with non-trivial bundle, which is closely related to the kernel of $u^2-64$. We discuss these results in the context of the moduli space of stable rank two holomorphic bundles with fixed odd determinant over a Riemann surface.