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Constraints on embedded spheres and real projective planes in 4-manifolds from Seiberg-Witten theory

David Baraglia

Abstract

We calculate the Seiberg-Witten invariants of branched covers of prime degree, where the branch locus consists of embedded spheres. Aside from the formula itself, our calculations give rise to some new constraints on configurations of embedded spheres in 4-manifolds. Using similar methods, we also obtain new constraints on embeddings of real projective planes and spheres with a cusp singularity. Moreover, we show that the existence of certain configurations of surfaces would give rise to 4-manifolds of non-simple type. Our proof makes use of equivariant Seiberg-Witten invariants as well as a gluing formula for the relative Seiberg-Witten invariants of 4-manifolds with positive scalar curvature boundary.

Constraints on embedded spheres and real projective planes in 4-manifolds from Seiberg-Witten theory

Abstract

We calculate the Seiberg-Witten invariants of branched covers of prime degree, where the branch locus consists of embedded spheres. Aside from the formula itself, our calculations give rise to some new constraints on configurations of embedded spheres in 4-manifolds. Using similar methods, we also obtain new constraints on embeddings of real projective planes and spheres with a cusp singularity. Moreover, we show that the existence of certain configurations of surfaces would give rise to 4-manifolds of non-simple type. Our proof makes use of equivariant Seiberg-Witten invariants as well as a gluing formula for the relative Seiberg-Witten invariants of 4-manifolds with positive scalar curvature boundary.

Paper Structure

This paper contains 22 sections, 48 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Configurations of embedded surfaces
  3. Seiberg--Witten invariants of branched covers
  4. Potential constructions of non-simple type $4$-manifolds
  5. Method of calculation
  6. Structure of paper
  7. Spherical $3$-manifolds
  8. Lens spaces
  9. Prism manifolds
  10. Plumbing on $ADE$ diagrams
  11. Circle bundles over $\mathbb{RP}^2$
  12. Seiberg--Witten Floer homotopy type
  13. Seiberg--Witten invariants for $4$-manifolds with psc boundary
  14. Equivariant and reduced Seiberg--Witten invariants
  15. An adjunction-type inequality
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $p$ be a prime number. Let $X$ be a compact, oriented, smooth $4$-manifold with $H_1(X ; \mathbb{Z}_p) = 0$ and $b_+(X) > 1$. Suppose $X$ is of simple type and let $\mathfrak{s}$ be a spin$^c$-structure such that $SW(X , \mathfrak{s}) \neq 0 \; ({\rm mod} \; p)$. Let $S \subset X$ be a smoothly

Theorems & Definitions (87)

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