3
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.