In relation to the –dimensional smooth Poincaré conjecture, we construct a tentative invariant of homotopy –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) –spheres.
Chris Gerig. "No homotopy –sphere invariants using ." Algebr. Geom. Topol. 21 (5) 2543 - 2569, 2021. https://doi.org/10.2140/agt.2021.21.2543