| Embedded contact homology (abbreviated ECH) is a topological invariant of a contact manifold (Y,Ker lambda) which is defined in terms of a contact form lambda. It was introduced by Michael Hutchings in [5], [9]. The key nontrivial part of the definition of ECH is the ECH index. The ECH differential ∂ counts embedded pseudoholomorphic curves . more precisely, those with ECH index 1 -- in the symplectization of Y . Hutchings and Taubes [7] later proved that ∂ indeed satisfies ∂ 2 = 0. Taubes [10] also proved that ECH is, in fact, isomorphic to Seiberg-Witten Floer cohomology. Thus, ECH is independent of the choice of the contact structure, contact form, and almost complex structure J.;In this paper, we will construct a combinatorial version of ECH for the unit cotangent bundle S1T*S of a pair-of-pants S. Using the results of Cieliebak-Latschev from [1], we are able to reduce the ECH differential to a count of certain string operations (Goldman brackets [3],[11]) on immersed closed geodesics on S. We will give a combinatorial proof of ∂2 = 0 with this model, as an alternative to Hutchings and Taubes' proof. The combinatorial construction, with the aid of a computer, allows us to compute ECH( S1T*S, lambda, J;mGamma A + nGammaB), for certain classes GammaA, GammaB ∈ H 1(S1T*S). Moreover, we conjecture that ECH(S1 T*S, lambda, J;mGammaA + nGammaB) = 0 for all m, n > 0, where m + n > 1. |
