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arXiv · 1905.10938

No homotopy 4-sphere invariants using ECH = SWF

Chris Gerig
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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.

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Chris Gerig. 2019-05-27. No homotopy 4-sphere invariants using ECH = SWF. https://doi.org/10.2140/agt.2021.21.2543

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