Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them.

In the symplectic world, there are sum formulas for Gromov-Witten invariants of symplectic manfiolds which can be described a symplectic sum of two symplectic manifolds.

Are there any similar decompositions of contact 3-manifolds which lead to interesting relations in ECH similar to the Gromov-Witten sum formula?


Yes, given two contact 3-manifolds $(M_1,\xi_1)$ and $(M_2,\xi_2)$ we can form their contact sum $(M_1\# M_2,\xi_1\# \xi_2)$ and then ECH decomposes as the tensor product of the corresponding ECH's of the pieces (at least assuming field coefficients).

See the paper Sutures and Contact Homology by Colin-Ghiggini-Honda-Hutchings, in particular Theorem 1.8 and its proof in Section 8. It also contains results on sutured manifold decompositions and how ECH respects that -- the maps on ECH under "sutured manifold gluing" are monomorphisms.

I will also use this space to point out that $ECH$, as opposed to the other isomorphic homologies $HF$ and $SWF$, easily handles disconnected manifolds. Roughly, ECH of the disjoint union of two contact 3-manifolds is the tensor product of their ECH's.


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