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Seiberg-Witten Floer spectra for $b_1>0$

Hirofumi Sasahira, Matthew Stoffregen

Abstract

We construct a generalization of the Seiberg-Witten Floer spectrum for suitable three-manifolds $Y$ with $b_1(Y)>0$. For a cobordism between three-manifolds we define Bauer-Furuta maps on these new spectra, and additionally compute some examples.

Seiberg-Witten Floer spectra for $b_1>0$

Abstract

We construct a generalization of the Seiberg-Witten Floer spectrum for suitable three-manifolds with . For a cobordism between three-manifolds we define Bauer-Furuta maps on these new spectra, and additionally compute some examples.

Paper Structure

This paper contains 36 sections, 98 theorems, 684 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Finite-dimensional approximation
  4. Four-manifolds
  5. Further Directions
  6. Organization
  7. Finite dimensional approximation on 3-manifolds
  8. Spectral sections
  9. Connections on Hilbert bundles
  10. Notation and main statements
  11. Construction of spectral sections
  12. Derivative of projections
  13. Weighted Sobolev space
  14. Proof of Theorem \ref{['thm isolating nbd']}
  15. Well-definedness
  16. ...and 21 more sections

Key Result

Theorem 1.1.1

Let $(Y,\mathfrak{s})$ be a closed, $\mathrm{spin}^c$$3$-manifold which satisfies that the first Chern class $c_1(\mathfrak{s})\in H^2(Y;\mathbb{Z})$ is torsion, and so that the triple-cup product on $H^1(Y;\mathbb{Z})$ vanishes. Associated to a Floer framing$\mathfrak{P}$ (see Section subsec:defn-o for some $n\in \mathbb{Z}$, depending only on $\mathfrak{P}$.

Theorems & Definitions (209)

  • Theorem 1.1.1
  • Theorem 1.2.1
  • Conjecture 1.2.2
  • Theorem 1.3.1
  • Theorem 1.3.2
  • Theorem 1.3.3
  • Definition 2.1.1: MP
  • Theorem 2.1.2
  • Proposition 2.1.3
  • Definition 2.3.1
  • ...and 199 more