# No homotopy 4-sphere invariants using ECH = SWF

## TL;DR

The paper attempts to construct a homotopy 4-sphere invariant using ECH and SWF but finds it to be constant, indicating these theories are unsuitable for this purpose, yet reveals existence of special pseudoholomorphic curves.

## Contribution

It introduces a tentative invariant for homotopy 4-spheres via ECH and SWF and demonstrates its triviality, guiding future research directions.

## Key findings

- The invariant is constant across all homotopy 4-spheres.
- ECH and SWF are not effective for distinguishing homotopy 4-spheres.
- Existence of pseudoholomorphic curves with specific constraints in 4-spheres.

## Abstract

In relation to the 4-dimensional smooth Poincar\'e 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.