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

Cobordism maps in embedded contact homology