Cobordism maps in embedded contact homology

Abstract

Given an exact symplectic cobordism $(X,\lambda )$ between contact $3$-manifolds $(Y_{+},{\lambda }_{+})$ and $(Y_{-},{\lambda }_{-})$ with no elliptic Reeb orbits up to a certain action, we define a chain map from the embedded contact homology (ECH) chain complex of $(Y_{+},{\lambda }_{+})$ to that of $(Y_{-},{\lambda }_{-})$, both taken with coefficients in $\mathbb{Z}/2\mathbb{Z}$. The map is defined by counting punctured holomorphic curves with ECH index $0$ in the completion of the cobordism and new objects that we call ECH buildings, answering a question of Hutchings.

64 pages; minor edits

References in corpus (1)

• Semi-global Kuranishi charts and the definition of contact homology

Cited by in corpus (3)

• On cobordism maps on periodic Floer homology
• PFH spectral invariants and $C^{\infty }$ closing lemmas
• Combinatorial Reeb dynamics on punctured contact 3-manifolds