[Paper Review] Floer Homology, Nielsen Theory and Symplectic Zeta Functions

Alexander Fel’shtyn|ArXiv.org|Nov 2, 2002

Geometric and Algebraic Topology3 citations

TL;DR

This paper establishes a deep connection between symplectic Floer homology and Nielsen fixed point theory on surfaces, introducing symplectic zeta functions and an asymptotic invariant derived from Floer homology growth. It proves that for monotone symplectomorphisms, the dimension of symplectic Floer homology equals the Nielsen number, and special values of the zeta functions coincide with Reidemeister torsions, linking topological invariants to symplectic geometry.

ABSTRACT

We describe a connection between symplectic Floer homology for symplectomorphisms of surface and Nielsen fixed point theory. A new zeta functions and asymptotic invariant of symplectic origin are defined. We show that special values of symplectic zeta functions are Reidemeister torsions.

Motivation & Objective

• To establish a correspondence between symplectic Floer homology and Nielsen fixed point theory on surfaces.
• To define new symplectic zeta functions and an asymptotic invariant based on the growth of Floer homology dimensions.
• To prove that special values of these zeta functions are equal to Reidemeister torsions, linking symplectic and topological invariants.
• To extend the framework to general mapping classes, particularly pseudo-Anosov and finite-type diffeomorphisms, via conjectures on monotone representatives.
• To propose that symplectic zeta functions and the asymptotic invariant yield new invariants for contact 3-manifolds and symplectic 4-manifolds.

Proposed method

• Define symplectic zeta functions using the growth of Nielsen numbers and Floer homology dimensions.
• Use the monotonicity condition on symplectomorphisms to ensure well-defined Floer homology and relate it to topological invariants.
• Apply the short exact sequence in cohomology to characterize monotonicity via the class $ m( heta) $, with $ m( heta) = 0 $ implying monotonicity.
• Relate the dimension of symplectic Floer homology $ \dim HF_*(\phi) $ to the Nielsen number $ N(\phi) $, showing $ \dim HF_*(\phi) = N(\phi) $ for periodic and finite-type diffeomorphisms.
• Use Markov partitions and symbolic dynamics for pseudo-Anosov maps to derive a trace formula for Nielsen numbers via transition matrices.
• Conjecture that the symplectic zeta function is rational and given by $ F_g(z) = \prod_{i=0}^m \det(1 - A_i z)^{-\epsilon_i} $, based on the trace formula and combinatorial cancellations.

Experimental results

Research questions

• RQ1 How are symplectic Floer homology and Nielsen fixed point theory related on surfaces?
• RQ2 What is the analytical structure of the new symplectic zeta functions defined via Floer homology?
• RQ3 Can special values of these zeta functions be identified with known topological invariants such as Reidemeister torsion?
• RQ4 What is the asymptotic growth of Floer homology dimensions, and how does it relate to topological entropy and the stretching factor of pseudo-Anosov maps?
• RQ5 Does every mapping class admit a monotone representative such that its Floer homology reflects the Nielsen number and asymptotic growth?

Key findings

• For non-trivial orientation-preserving periodic diffeomorphisms or finite-type diffeomorphisms with isolated fixed points, $ \dim HF_*(\phi) = N(\phi) $, and $ HF_*(\phi) \cong \mathbb{Z}_2^{N(\phi)} $.
• Special values of the symplectic zeta functions are equal to Reidemeister torsions, establishing a direct link between symplectic and topological invariants.
• The asymptotic invariant $ F^\infty(g) = \limsup_{n \to \infty} |N(\phi^n)|^{1/n} $ equals the topological entropy $ h(\psi) $ and the stretching factor $ \lambda $ of the Thurston canonical form $ \psi $.
• For pseudo-Anosov mapping classes, the symplectic zeta function is conjectured to be rational and given by $ F_g(z) = \prod_{i=0}^m \det(1 - A_i z)^{-\epsilon_i} $, based on Markov partition dynamics.
• The growth rate of $ \dim HF_*(\phi^n) $ matches the topological entropy $ h(\psi) $, and for pseudo-Anosov maps, $ F^\infty(g) = \lambda > 1 $.
• The dimension of Floer homology is a new symplectic invariant for 4-manifolds with non-zero first Betti number, as shown by Seidel, and this paper extends its interpretation via Nielsen theory.