Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

William Chen, Christopher Scaduto

Abstract

In the description of the instanton Floer homology of a surface times a circle due to Muñoz, 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.

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

Abstract

In the description of the instanton Floer homology of a surface times a circle due to Muñoz, we compute the nilpotency degree of the endomorphism . 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 . 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.

Paper Structure

This paper contains 10 sections, 21 theorems, 83 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The analogy with singular cohomology
  3. The ring structure of the instanton homology
  4. Computing the nilpotency degree
  5. The framed instanton homology
  6. The kernel of $\beta+8$.
  7. The kernel of $\beta-8$.
  8. The betti numbers
  9. Comparison to the cohomology of the critical set
  10. Appendix: the twisted Gysin sequence

Key Result

Theorem 1

The endomorphism $u^2-64$ acting on the instanton homology $I(\Sigma\times S^1)_w$ satisfies

Theorems & Definitions (32)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3: munoz-ring
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['prop:main']}
  • Lemma 3
  • ...and 22 more