Volume 21, issue 5 (2021)

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

Chris Gerig

Algebraic & Geometric Topology 21 (2021) 2543–2569

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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ECH, near-symplectic, 4–sphere, Seiberg–Witten, Gromov
Mathematical Subject Classification
Primary: 53D42
Secondary: 57K41
Received: 13 April 2020
Revised: 11 October 2020
Accepted: 27 October 2020
Published: 31 October 2021
Chris Gerig
Mathematics Department
Harvard University
Cambridge, MA
United States