[Paper Review] p-cyclic persistent homology and Hofer distance

Jun Zhang|arXiv (Cornell University)|May 24, 2016

Topological and Geometric Data Analysis12 references3 citations

TL;DR

This paper establishes that the Hofer distance between time-dependent Hamiltonian diffeomorphisms and $p$-th power Hamiltonian diffeomorphisms on $\Sigma_g \times M$ can be arbitrarily large when $g \geq 4$ and $p$ is sufficiently large. Using $p$-cyclic persistent homology and singular value decomposition in a Floer-theoretic setting with group actions, the authors generalize Polterovich and Shelukhin's result, proving $\text{power}_p(\Sigma_g \times M) = \infty$, which implies $\text{aut}(\Sigma_g \times M) = \infty$, supporting the conjecture that $\text{aut}(X) = \infty$ for all closed symplectic manifolds.

ABSTRACT

In this paper, we generalize the result from L. Polterovich and E. Shelukhin's paper stating that Hofer distance from time-dependent Hamiltonian diffeomorphism to the set of p-th power Hamiltonian diffeomorphism can be arbitrarily large to hold in the product structure $Σ_g imes M$ for any closed symplectic manifold $M$ when $p$ is sufficiently large and $g \geq 4$. This implies that, on this product, Hofer distance can be arbitrarily large between time-dependent Hamiltonian diffeomorphism and autonomous Hamiltonian diffeomorphism.The basic tool we use is barcode and singular value decomposition that are developed in previous joint work with M. Usher, from which we borrow many proofs and modify them so that it can be adapted to the situation that filtered chain complex equipped with a group action.

Motivation & Objective

• To extend Polterovich and Shelukhin's result on Hofer distance in Hamiltonian dynamics to a broader class of symplectic manifolds.
• To establish that $\text{power}_p(\Sigma_g \times M) = \infty$ for $g \geq 4$ and large prime $p$, implying $\text{aut}(\Sigma_g \times M) = \infty$.
• To develop and apply $p$-cyclic persistent homology techniques in a Floer-theoretic setting with group actions, generalizing previous methods.
• To provide a homological obstruction using barcode multiplicity and singular value decomposition to detect non-trivial Hofer distance.

Proposed method

• Construct a $p$-cyclic action on the Floer complex via rotation of loops, inducing a $\mathbb{Z}/p\mathbb{Z}$-action on the chain level.
• Apply $p$-cyclic singular value decomposition to analyze the structure of the filtered Floer complex under this group action.
• Use barcode decomposition of the mapping cone of the $p$-fold power map to extract invariants related to Hofer distance.
• Leverage the product structure of barcodes on $\Sigma_g \times M$ to relate the barcode of the product to those of the factors.
• Compute the multiplicity of finite-length bars in the barcode of the product complex using binomial coefficients and Babbage's theorem.
• Analyze divisibility of barcode multiplicity modulo $p$ to detect non-triviality of the Hofer distance.

Experimental results

Research questions

• RQ1 Can the Hofer distance between time-dependent Hamiltonian diffeomorphisms and $p$-th power Hamiltonian diffeomorphisms be arbitrarily large on $\Sigma_g \times M$ for $g \geq 4$?
• RQ2 Does the $p$-cyclic persistent homology framework provide a robust obstruction to the existence of $p$-th roots in the Hamiltonian group?
• RQ3 How does the barcode structure of the Floer complex transform under the $p$-fold power map and group action?
• RQ4 What is the role of the egg-beater model in computing barcode invariants for $\Sigma_g$ with $g \geq 4$?
• RQ5 Under what conditions is the multiplicity of degree-1 bars in the product complex not divisible by $p$, implying non-trivial Hofer distance?

Key findings

• The Hofer distance $\text{power}_p(\Sigma_g \times M)$ is infinite for $g \geq 4$ and sufficiently large prime $p$, generalizing the result of Polterovich and Shelukhin.
• The multiplicity of degree-1 finite-length bars in the barcode of the product complex $\Sigma_g \times M$ is given by $\sum_{k=-p+1}^{p+1} \binom{2p}{k+p-1} \cdot qb_{1-k}(M)$, which is not divisible by $p$ when $p > 2\sum_{0 \leq i \leq 2n} b_i(M)$.
• The middle term $\binom{2p}{p} \cdot qb_0(M)$ in the multiplicity sum satisfies $\binom{2p}{p} \equiv 2 \pmod{p}$, contributing a non-zero term modulo $p$.
• When $M = \mathbb{C}P^n$, the multiplicity modulo $p$ is $2$, so non-divisibility holds for all primes $p \geq 3$, implying $\text{power}_p(\Sigma_g \times \mathbb{C}P^n) = \infty$.
• For $c_1(TM) = 0$, the multiplicity modulo $p$ is $b_p(M) + 2$, so non-divisibility holds if $p \nmid b_p(M) + 2$, providing a refined obstruction condition.
• The construction of $p$-cyclic persistent homology via singular value decomposition allows a homological obstruction to $p$-th roots in the Hamiltonian group, enabling the proof of unbounded Hofer distance.