Seiberg-Witten and Gromov invariants for self-dual harmonic 2-forms

TL;DR

This paper extends the Seiberg-Witten and Gromov invariants relationship to non-symplectic 4-manifolds, showing their equivalence modulo 2 in the presence of self-dual harmonic 2-forms, and discusses implications for 3-manifold invariants.

Contribution

It generalizes Taubes' SW=Gr theorem to non-symplectic 4-manifolds and establishes their invariants' equivalence under broader conditions.

Findings

Seiberg-Witten invariants are equivalent modulo 2 to Gromov invariants on certain 4-manifolds.

Extension of results to non-minimal 4-manifolds.

Connection to circle-valued Morse theory on 3-manifolds.

Abstract

This is the sequel to the author's previous paper which gives an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds. The main result of this paper asserts the following. Whenever the Seiberg-Witten invariants are defined over a closed minimal 4-manifold X, they are equivalent modulo 2 to "near-symplectic" Gromov invariants in the presence of certain self-dual harmonic 2-forms on X. A version for non-minimal 4-manifolds is also proved. A corollary to circle-valued Morse theory on 3-manifolds is also announced, recovering a result of Hutchings-Lee-Turaev about the 3-dimensional Seiberg-Witten invariants.