Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exotic embedded surfaces and involutions from Real Seiberg-Witten theory

David Baraglia

Abstract

Using Real Seiberg--Witten theory, Miyazawa introduced an invariant of certain 4-manifolds with involution and used this invariant to construct infinitely many exotic involutions on $\mathbb{CP}^2$ and infinitely many exotic smooth embeddings of $\mathbb{RP}^2$ in $S^4$. In this paper we extend Miyazawa's construction to a large class of 4-manifolds, giving many infinite families of involutions on 4-manifolds which are conjugate by homeomorphisms but not by diffeomorphisms and many infinite families of exotic embeddings of non-orientable surfaces in 4-manifolds, where exotic means continuously isotopic but not smoothly isotopic. Exoticness of our construction is detected using Real Seiberg--Witten theory. We study Miyazawa's invariant, relate it to the Real Seiberg--Witten invariants of Tian--Wang and prove various fundamental results concerning the Real Seiberg--Witten invariants such as: relation to positive scalar curvature, wall-crossing, a mod 2 formula for spin structures, a localisation formula relating ordinary and Real Seiberg--Witten invariants, a connected sum formula and a fibre sum formula.

Exotic embedded surfaces and involutions from Real Seiberg-Witten theory

Abstract

Using Real Seiberg--Witten theory, Miyazawa introduced an invariant of certain 4-manifolds with involution and used this invariant to construct infinitely many exotic involutions on and infinitely many exotic smooth embeddings of in . In this paper we extend Miyazawa's construction to a large class of 4-manifolds, giving many infinite families of involutions on 4-manifolds which are conjugate by homeomorphisms but not by diffeomorphisms and many infinite families of exotic embeddings of non-orientable surfaces in 4-manifolds, where exotic means continuously isotopic but not smoothly isotopic. Exoticness of our construction is detected using Real Seiberg--Witten theory. We study Miyazawa's invariant, relate it to the Real Seiberg--Witten invariants of Tian--Wang and prove various fundamental results concerning the Real Seiberg--Witten invariants such as: relation to positive scalar curvature, wall-crossing, a mod 2 formula for spin structures, a localisation formula relating ordinary and Real Seiberg--Witten invariants, a connected sum formula and a fibre sum formula.

Paper Structure

This paper contains 25 sections, 46 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Exotic involutions and exotic embedded surfaces
  3. Real Seiberg--Witten invariants
  4. Structure of the paper
  5. Real spin$^c$-structures
  6. Real Seiberg--Witten invariants
  7. Real Bauer--Furuta invariants
  8. $O(2)$-equivariant Bauer--Furuta map
  9. Real Bauer--Furuta map
  10. Integer invariants
  11. Integer Real Seiberg--Witten invariant
  12. Degree of the Real Bauer--Furuta map
  13. Properties of the Real Seiberg--Witten invariants
  14. Positive scalar curvature
  15. Wall-crossing formula
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $(X_1, \sigma_1), \dots , (X_k , \sigma_k)$ be admissible pairs. Let $c : \mathbb{CP}^2 \to \mathbb{CP}^2$ denote complex conjugation. Let $X = X_1 \# \cdots \# X_k$ and $\sigma = \sigma_1 \# \cdots \# \sigma_k$. Then

Theorems & Definitions (87)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 77 more