#### Volume 24, issue 4 (2020)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Taming the pseudoholomorphic beasts in $\mathbb{R} \times (S^1 \times S^2)$

### Chris Gerig

Geometry & Topology 24 (2020) 1791–1839
##### Abstract

For a closed oriented smooth $4$–manifold $X$ with ${b}_{+}^{2}\left(X\right)>0$, the Seiberg–Witten invariants are well-defined. Taubes’ “$SW=Gr$” theorem asserts that if $X$ carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes’ Gromov invariants. In the absence of a symplectic form, there are still nontrivial closed self-dual $2$–forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describe well-defined integral counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic $2$–forms, and it is shown that they recover the Seiberg–Witten invariants over $ℤ∕2ℤ$. This is an extension of “$SW=Gr$” to nonsymplectic $4$–manifolds.

The main result of this paper asserts the following. Given a suitable near-symplectic form $\omega$ and tubular neighborhood $\mathsc{𝒩}$ of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism $\left(X-\mathsc{𝒩},\omega \right)$ which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form “near-symplectic” Gromov invariants as a function of spin-c structures on $X\phantom{\rule{-0.17em}{0ex}}$.

##### Keywords
near-symplectic, Seiberg–Witten, Gromov, pseudoholomorphic, ECH
Primary: 53D42
Secondary: 57R57