Automatic transversality in contact homology II: filtrations and computations

Abstract

This paper is the sequel to the previous paper [Nelson, Abh. Math. Semin. Univ. Hambg. 85 (2015) , which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically convex contact forms. The Reeb orbits of these so-called dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder indices with respect to a free homotopy class. Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show how to obtain the desired cylindrical contact homology by taking direct limits. We give a direct proof of invariance of cylindrical contact homology within the class of dynamically separated contact forms, and elucidate the independence of the filtered cylindrical contact homology with respect to the choice of the dynamically separated contact form and almost complex structure. We also show that these regularity results are compatible with geometric methods of computing cylindrical contact homology of prequantization bundles, proving a conjecture of Eliashberg [Symplectic field theory and its applications, International Congress of Mathematicians I (European Mathematical Society, Zürich, 2007) 217-246] in dimension three.Definition 1.6. Let (M, λ) be a contact 3-manifold with c 1 (ker λ) = 0 such that all the Reeb orbits of R λ are contractible. Then λ is said to be dynamically separated whenever the following conditions hold.(I) If γ is a closed simple Reeb orbit then 3 μ CZ (γ) 5. (II) If γ k is the k-fold cover of a simple orbit γ then μ CZ (γ k ) = μ CZ (γ k−1 ) + 4. * (Y, λ, J).Corollary 1.18. CH EGH * is an invariant of closed contact manifolds (Y, ξ) for which there exists a pair (λ, J) where λ is a nondegenerate hypertight contact form with ker(λ) = ξ, and J is an admissible λ-compatible almost complex structure.Again, we will upgrade these results to hold for dynamically convex contact forms in dimension three in a forthcoming paper by Hutchings and Nelson. In contrast to [30] and a forthcoming paper by Hutchings and Nelson, this paper is concerned with the more restricted class of dynamically separated contact forms which allows us to directly obtain regularity for S 1 -independent pseudoholomorphic cylinders in cobordisms. Filtered cylindrical contact homologyThe action of a Reeb orbit γ is given by A(γ) := γ λ.. Since J is a λ-compatible almost complex structure on the symplectization it follows [40, Lemma 2.18] that the cylindrical contact homology differential(s) decreases the action, for example, ifThus, given any real number L it makes sense to define the filtered cylindrical contact homology, denoted by CH EGH,L * (M, λ, J), to be the homology of the subcomplex C EGH,L * (M, λ, J) of the chain complex spanned by generators of action less than L. The invariance of these filtered cylindrical contact homology groups is more subtle than in the unfiltered case, as they typically depend ...

Reeb dynamics of the link of the An singularity

“…Remark 1.7. Ongoing work of Nelson [18] and Hutchings and Nelson [8] is needed in order to work under the assumption that a related Floer-theoretic invariant, cylindrical contact homology, is a well-defined contact invariant of (L An , ξ 0 ). Once this is complete, the index calculations provided in Theorem 1.1 show that positive S 1 -equivariant symplectic homology and cylindrical contact homology agree up to a degree shift.…”

Cylindrical contact homology of links of simple singularities

“…Proof. That γ is nondegenerate and projects to a critical point p of f is proven in [N2,Lemma 4.11]. To compute µ CZ (γ), we apply the naturality, loop, and signature properties of the Conley Zehnder index (see [S, §2.4]) to our path {M t } ⊂ Sp(2).…”

Cylindrical contact homology of 3-dimensional Brieskorn manifolds

“…In this section we define a perturbation of λ p,q,r to give a nondegenerate contact form on Σ(p, q, r), which we can then use to compute contact homology. Strictly, our construction gives a contact form that is nondegenerate "up to large action" in the sense of Nelson [9], but that will be sufficient for our purposes.…”

“…This is geometrically pretty, but of course the form then fails to be nondegenerate. Our coordinate description allows us to perturb the form to achieve nondegeneracy (up to a fixed action, at least), following ideas in [9]. This is done in section 4, where the crucial calculation is the Conley-Zehnder indices of the resulting nondegenerate closed orbits.…”

“…It is this calculation that determines the grading on cylindrical contact homology, and the perturbation and index calculations are the essentially new contributions of our work. The differential on the complex is mostly trivial for grading reasons, but there are nonzero differentials that we compute by reducing to the situation of [9].…”