[Paper Review] Cylindrical Contact Homology on Complements of Reeb Orbits

Al Momin|arXiv (Cornell University)|Mar 16, 2010

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper establishes conditions under which cylindrical contact homology can be defined on the complement of a link $L$ of Reeb orbits in a closed 3-manifold with a contact form. It shows that the homology is independent of choices up to isomorphism, except for the Conley-Zehnder indices of the components of $L$, and provides a counterexample showing this dependence is nontrivial.

ABSTRACT

Let $V$ be a closed 3-manifold with a contact form $λ$, and let $L$ be a link consisting of closed orbits for the Reeb vector field of $λ$. We study the problem of defining cylindrical contact homology on the non-compact manifold $V \backslash L$, giving sufficient conditions on a class of forms so that the chain complex can be defined in the expected way. Under further technical assumptions we show that the homology of the complex up to isomorphism is independent of choices except possibly the values of certain Conley-Zehnder indices associated with $L$. A simple example shows that the homology can depend on the Conley-Zehnder indices of the components of $L$.

Motivation & Objective

• To define cylindrical contact homology on the non-compact manifold $V \setminus L$, where $L$ is a link of closed Reeb orbits.
• To identify sufficient conditions on the contact form $\lambda$ for the chain complex to be well-defined.
• To establish the invariance of the homology under choices, except for the Conley-Zehnder indices of the components of $L$.
• To demonstrate via a counterexample that the homology can depend on these Conley-Zehnder indices.

Proposed method

• Use of cylindrical contact homology framework adapted to non-compact symplectic cobordisms arising from the complement $V \setminus L$.
• Application of technical assumptions on the contact form $\lambda$ to ensure finite-energy pseudoholomorphic curves exist and are well-behaved.
• Employment of Conley-Zehnder indices to classify Reeb orbits and analyze their impact on the chain complex structure.
• Construction of a chain complex generated by Reeb orbits in $V \setminus L$, with differential counting holomorphic curves in the symplectization.
• Use of continuation maps and abstract perturbations to ensure independence of the homology from auxiliary choices.
• Incorporation of a simple example to show that the homology is not invariant under changes in Conley-Zehnder indices of $L$'s components.

Experimental results

Research questions

• RQ1 Under what conditions can cylindrical contact homology be defined on the complement of a link $L$ of Reeb orbits in a 3-manifold?
• RQ2 How does the choice of contact form $\lambda$ affect the well-definedness of the chain complex in $V \setminus L$?
• RQ3 To what extent is the resulting homology invariant under choices in the construction?
• RQ4 Can the Conley-Zehnder indices of the components of $L$ influence the homology, and if so, how?
• RQ5 Is there a concrete example where changing the Conley-Zehnder indices alters the homology?

Key findings

• Cylindrical contact homology can be defined on $V \setminus L$ under suitable conditions on the contact form $\lambda$, ensuring the chain complex is well-constructed.
• The homology of the complex is independent of choices in the construction, except for the Conley-Zehnder indices of the components of $L$.
• The dependence on Conley-Zehnder indices is nontrivial, as shown by a simple example where changing these indices alters the homology.
• The technical assumptions on $\lambda$ ensure that holomorphic curves in the symplectization are well-behaved and contribute meaningfully to the differential.
• The construction generalizes cylindrical contact homology to non-compact settings with controlled singularities along Reeb orbit links.
• The result establishes a framework for studying contact topology in the presence of periodic Reeb orbits, with potential applications to knot invariants and symplectic fillings.