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$Pin^{-}(2)$ Bauer-Furuta invariants

Hao Wu

Abstract

Adapting Bauer and Furuta's constructions of the refinement of the Seiberg-Witten invariants, we establish the analogous stable cohomotopy refinement of the $Pin^{-}(2)$ monopole invariants proposed by Nakamura \cite{nakamura2015pin}, and give the corresponding connected sum formula.

$Pin^{-}(2)$ Bauer-Furuta invariants

Abstract

Adapting Bauer and Furuta's constructions of the refinement of the Seiberg-Witten invariants, we establish the analogous stable cohomotopy refinement of the monopole invariants proposed by Nakamura \cite{nakamura2015pin}, and give the corresponding connected sum formula.

Paper Structure

This paper contains 19 sections, 21 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $Spin^{c-}$ structures
  4. Spinor bundles and Clifford multiplication
  5. Dirac operators
  6. $Pin^{-}(2)$ monopole equations
  7. $Pin^{-}(2)$ monopole invariants
  8. A gluing formula
  9. The monopole map and the stable cohomotopy invariant
  10. Connected sum formula
  11. The setup
  12. Proof of Theorem \ref{['main theorem2']}
  13. Proof of Theorem \ref{['gluing isomorphism']}
  14. The standard estimates
  15. Varying the length of the long neck
  16. ...and 4 more sections

Key Result

Theorem 1.1

The monopole map $\mu:\mathcal{A}\to \mathcal{C}$ defines an element in an equivariant stable cohomotopy group which is independent of the chosen Riemannian metric. For $b^+(X;l)>\dim(Pic^c(X))+1$, we have a natural homomorphism of the cohomotopy group to $\mathbb{Z}_2$, which maps $[\mu]$ to the $\mathbb{Z}_2$-valued $Pin^{-}(2)$ monopole invariant.

Figures (2)

  • Figure 1: homotopy commutative diagram
  • Figure 2: join commutative diagram

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: $Spin^{c-}$ groups
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Clifford multiplication $1$
  • Definition 2.7: Clifford multiplication $2$
  • ...and 43 more