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Extending the flux homomorphism to volume-preserving homeomorphisms

Stéphane Tchuiaga

TL;DR

The work develops a $C^0$-robust extension of the flux homomorphism to volume-preserving homeomorphisms by introducing gap functions $\mathbf{R}_\delta(H,\alpha)$ and an extended flux $\widetilde{L}_\Omega^\delta$, establishing a precise factorization with the smooth flux and a Poincaré-duality framework via Fathi’s mass flow. A central result is the $(C^0,\delta)$-rigidity: the extended flux group $\tilde{\Gamma}_\Omega^\delta$ coincides with the classical flux group $\Gamma_\Omega$, indicating that small $C^0$ perturbations do not enlarge the flux possibilities. The paper also develops a cohomological perspective for $Homeo_0(M,\Omega)$, showing a natural injection of $H^1(M,\mathbb{R})$ into $H^1(Homeo_0(M,\Omega),\mathcal{C}(M,\mathbb{R}))$ and a lower bound on the first cohomology, thereby linking topology to dynamical constraints. Collectively, these results bridge smooth and topological dynamics for volume-preserving maps, propose a norm measuring form-distortion under homeomorphisms, and lay groundwork for rigidity phenomena and energy considerations in symplectic topology and related conservative systems.

Abstract

This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising $(C^0, δ)-$rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincaré duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic homeomorphism of $(T^2, ω)$ with trivial flux, is a finite energy Hamiltonian homeomorphism of $(T^2, ω)$. We discuss the cohomology groups $H^\ast(Homeo_0(M,Ω), \mathcal{C}(M, \mathbb{R}) )$ of $ Homeo_0(M,Ω)$ with coefficients in $ \mathcal{C}(M, \mathbb{R})$.

Extending the flux homomorphism to volume-preserving homeomorphisms

TL;DR

The work develops a -robust extension of the flux homomorphism to volume-preserving homeomorphisms by introducing gap functions and an extended flux , establishing a precise factorization with the smooth flux and a Poincaré-duality framework via Fathi’s mass flow. A central result is the -rigidity: the extended flux group coincides with the classical flux group , indicating that small perturbations do not enlarge the flux possibilities. The paper also develops a cohomological perspective for , showing a natural injection of into and a lower bound on the first cohomology, thereby linking topology to dynamical constraints. Collectively, these results bridge smooth and topological dynamics for volume-preserving maps, propose a norm measuring form-distortion under homeomorphisms, and lay groundwork for rigidity phenomena and energy considerations in symplectic topology and related conservative systems.

Abstract

This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincaré duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic homeomorphism of with trivial flux, is a finite energy Hamiltonian homeomorphism of . We discuss the cohomology groups of with coefficients in .
Paper Structure (10 sections, 27 theorems, 95 equations, 2 figures)

This paper contains 10 sections, 27 theorems, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Prelimimaries
  3. The gap functions: $\mathbf{R}_\delta(H, \alpha)$
  4. Properties of the functions $\mathbf{R}_\delta(H, \alpha)$
  5. Volume-preserving homeomorphisms
  6. On the kernel of $\ker L _\Omega^\delta$
  7. Measure-preserving homeomorphisms
  8. Basics notions
  9. Cohomology groups of $Homeo_0(M,\Omega)$ with coefficients in $\mathcal{C}(M,\mathbb{R})$
  10. Final remarks

Key Result

Theorem 1.1

Mull-1J-C Let $M$ be a closed connected smooth manifold of dimension $n$. If $n\leq 3$, then any homeomorphism can be uniformly approximated by diffeomorphisms. If $n\geq 5$, then a homeomorphism $h$ of $M$ can be uniformly approximated by a diffeomorphism $\phi$ if and only if $h$ is isotopic to a

Figures (2)

  • Figure 1: This image shows a smooth path $\Phi_t$ (blue) and a non-smooth path $H_t$ (red) connecting two points. Several geodesics $\chi_\delta^p$ connect points on both paths, illustrating how the gap is measured. A close up shows the correction term.
  • Figure 2: A sphere that represents the set $\mathcal{B}(1)$, and several graphs showing the values of the function $\chi(h,\alpha)(z)$ for different values of $\alpha$ in $\mathcal{B}(1)$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • ...and 37 more