#### Volume 21, issue 5 (2021)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
No homotopy $4$–sphere invariants using $\mathrm{ECH} = \mathrm{SWF}$

### Chris Gerig

Algebraic & Geometric Topology 21 (2021) 2543–2569
##### Abstract

In relation to the $4$–dimensional smooth Poincaré conjecture, we construct a tentative invariant of homotopy $4$–spheres using embedded contact homology (ECH) and Seiberg–Witten theory (SWF). But, for good reason, it is a constant value independent of the sphere, so this null result demonstrates that one should not try to use the usual theories of ECH and SWF. On the other hand, a corollary is that there always exist pseudoholomorphic curves satisfying certain constraints in (punctured) $4$–spheres.

##### Keywords
ECH, near-symplectic, 4–sphere, Seiberg–Witten, Gromov
Primary: 53D42
Secondary: 57K41