No homotopy 4–sphere invariants using ECH = SWF

Gerig, Chris

doi:10.2140/agt.2021.21.2543

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No homotopy $4$–sphere invariants using $\mathrm{ECH} = \mathrm{SWF}$

Chris Gerig

Algebraic & Geometric Topology 21 (2021) 2543–2569

DOI: 10.2140/agt.2021.21.2543

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.

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Keywords

ECH, near-symplectic, 4–sphere, Seiberg–Witten, Gromov

Mathematical Subject Classification

Primary: 53D42

Secondary: 57K41

References

Publication

Received: 13 April 2020

Revised: 11 October 2020

Accepted: 27 October 2020

Published: 31 October 2021

Authors

Chris Gerig
Mathematics Department Harvard University Cambridge, MA United States

Volume 21, issue 5 (2021)