Direct construction of ECH cobordism maps

Develop a direct definition of the ECH cobordism map induced by a weakly exact symplectic cobordism without using the isomorphism between embedded contact homology and Seiberg–Witten Floer homology.

Background

The paper uses canonical ECH cobordism maps associated with weakly exact symplectic cobordisms. These maps are known through the identification of embedded contact homology with Seiberg–Witten Floer homology, but the paper notes that a direct holomorphic-curve-based construction is not available. Establishing such a construction would provide an intrinsic definition of the maps within embedded contact homology and clarify their chain-level geometric content.

References

It is interesting to note that the this map was calculated only indirectly by using the isomorphism between ECH and Seiberg-Witten Floer Homology as in . It is an open problem to define this map without this isomorphism.

— Obstruction to Symplectic Embeddings between Toric Domains  (2608.27905 - Miranda et al., 28 Aug 2026) in Section 4, subsection “Cobordism Maps”