An alternative definition of cobordism map of ECH

TL;DR

This paper redefines the cobordism map in embedded contact homology (ECH) using holomorphic curves, proving its independence from complex structures and demonstrating equivalence with the traditional definition.

Contribution

It introduces an alternative formulation of the ECH cobordism map based on counting zero ECH index curves, linking it to Seiberg Witten theory.

Findings

Proves independence of the cobordism map from almost complex structures.

Shows the new definition is equivalent to the existing one.

Establishes a connection between ECH and Seiberg Witten theory.

Abstract

In this article, we reformulate the cobordism map of embedded contact homology, which is induced by exact symplectic cobordism and defined as direct limit of homomorphisms called filtered ECH cobordism map. The filtered ECH cobordism map is defined by counting embedded holomorphic curves with zero ECH index and we prove that it is independent on almost complex structure by Seiberg Witten theory. Moreover, our definition in fact is equivalent to the existing definition.