No homotopy 4-sphere invariants using ECH = SWF

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.

19 pages, to appear in Algebr. Geom. Topol. with grammatical edits

References in corpus (3)

• Vanishing of the contact homology of overtwisted contact 3--manifolds
• Taming the pseudoholomorphic beasts in $\mathbb{R}\times (S^1\times S^2)$
• Group actions, corks and exotic smoothings of R^4

Cited by in corpus (1)

• A Bauer-Furuta type refinement of Kronheimer-Mrowka's invariant for 4-manifolds with contact boundary