[Paper Review] Contact spectral invariants and persistence

Maia Fraser|arXiv (Cornell University)|Feb 20, 2015

Topological and Geometric Data Analysis8 references3 citations

TL;DR

This paper establishes a connection between contact spectral invariants and persistent homology by interpreting generating-function-based capacities as persistences of homology classes in a persistence module. It introduces a ${\mathbb{Z}}_k$-equivariant contact capacity that proves lens space orderability, offering a new perspective via persistent homology that aligns with existing results from contact homology and equivariant generating function homology.

ABSTRACT

This sketch shows that the usual generating function based capacities have an interpretation in the language of persistent homology as persistences of certain homology classes in the persistence module formed by the corresponding generating function homology groups. This viewpoint suggests various new invariants, in particular a $\mathbb{Z}_k$-equivariant capacity which can be used to prove orderability of lens spaces, proved by Milin (2008) using contact homology and by Sandon (2010) using equivariant generating function homology. These are informal notes originally circulated in January 2014.

Motivation & Objective

• To interpret standard generating-function-based symplectic and contact capacities within the language of persistent homology.
• To introduce a ${\mathbb{Z}}_k$-equivariant contact capacity as a new invariant using persistent homology.
• To demonstrate that this equivariant capacity can be used to prove orderability of lens spaces, matching results from contact homology and equivariant GF homology.
• To unify the framework of generating function capacities with persistent homology, especially in equivariant settings.
• To provide a conceptual bridge between topological data analysis and contact/symplectic geometry through persistence modules.

Proposed method

• The paper constructs a persistence module $V_a = H_p(E, E^a)$ from the sub-level sets of a generating function $S$ on a vector bundle $E$ over $M = S^{2n}$ or $S^{2n} \times S^1$, where $E^a = \{S \leq a\}$.
• It defines the spectral invariant $c(\phi)$ as the persistence of the class $\eta = \theta(\mu) \in H_p(E, E^{-\infty})$, where $\theta$ is the Thom isomorphism and $\mu$ is the orientation class of the base.
• For the ${\mathbb{Z}}_k$-equivariant case, it defines a persistence module $W_a(\phi)$ using equivariant homology groups $G_{{\mathbb{Z}}_k,p}^{(a,\infty]}(\phi)$, forming a filtered family of vector spaces with inclusion-induced maps.
• It proves that the ${\mathbb{Z}}_k$-equivariant capacity $c^p_{{\mathbb{Z}}_k}(\phi)$, defined as the persistence of generator $\eta^p$, is conjugation-invariant and monotone.
• It uses Sandon’s computation of $W_a$ for critical values to show $[c^p_{{\mathbb{Z}}_k}(\widehat{B(R)})] = [\ell R]$ when $p = 2n\ell$, confirming non-triviality of the invariant.
• It establishes that the contact capacity $c^p_{{\mathbb{Z}}_k,\text{contact}}(\widehat{\mathcal{U}})$ equals the symplectic capacity $c^p_{{\mathbb{Z}}_k,\text{symp}}(\mathcal{U})$ for $\widehat{\mathcal{U}} = \mathcal{U} \times S^1$, linking the two settings.

Experimental results

Research questions

• RQ1 How can standard generating-function-based capacities in contact and symplectic geometry be interpreted in terms of persistent homology?
• RQ2 What new invariants arise when extending the persistence framework to equivariant settings, particularly ${\mathbb{Z}}_k$-equivariance?
• RQ3 Can the ${\mathbb{Z}}_k$-equivariant capacity be used to prove non-squeezing theorems and lens space orderability?
• RQ4 How do the persistence modules formed by equivariant generating function homology groups behave under inclusion and filtration?
• RQ5 To what extent do the functorial properties of equivariant homology ensure invariance and monotonicity of the resulting capacities?

Key findings

• The standard contact and symplectic capacities defined via generating functions are equivalent to the persistence of specific homology classes in the persistence module $H_p(E, E^a)$.
• A new ${\mathbb{Z}}_k$-equivariant contact capacity $c^p_{{\mathbb{Z}}_k}(\phi)$ is defined as the persistence of the class $\eta^p$ in the equivariant homology group $W_a(\phi)$, which is conjugation-invariant and monotone.
• For a ball $\widehat{B(R)} \subset \mathbb{R}^{2n} \times S^1$, the ${\mathbb{Z}}_k$-equivariant capacity satisfies $[c^p_{{\mathbb{Z}}_k}(\widehat{B(R)})] = [\ell R]$ when $p = 2n\ell$, confirming its non-triviality.
• This capacity provides a new, persistence-theoretic proof of lens space orderability, matching results previously obtained via contact homology and equivariant GF homology.
• The contact capacity $c^p_{{\mathbb{Z}}_k,\text{contact}}(\widehat{\mathcal{U}})$ equals the symplectic capacity $c^p_{{\mathbb{Z}}_k,\text{symp}}(\mathcal{U})$ for $\widehat{\mathcal{U}} = \mathcal{U} \times S^1$, establishing consistency between symplectic and contact settings.