No homotopy 4-sphere invariants using ECH = SWF
Symplectic Geometry
2021-11-10 v3 Differential Geometry
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.
Cite
@article{arxiv.1905.10938,
title = {No homotopy 4-sphere invariants using ECH = SWF},
author = {Chris Gerig},
journal= {arXiv preprint arXiv:1905.10938},
year = {2021}
}
Comments
19 pages, to appear in Algebr. Geom. Topol. with grammatical edits