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Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism

Morimichi Kawasaki, Mitsuaki Kimura, Shuhei Maruyama, Takahiro Matsushita, Masato Mimura

TL;DR

The paper develops a general method to extract an invariant ℝ-valued bilinear form from a Symp^c(M,ω)–invariant homogeneous quasimorphism μ on the commutator subgroup N=[G,G], constructing 𝔟_μ on the abelianization Γ=G/N via a limit formula. Specializing to G=̃Symp^c(M,ω) and N=̃Ham^c(M,ω) with μ=μ_Sh, the authors prove the existence of an 𝔟_{μ_Sh} that is Symp^c(M,ω)–invariant and satisfies μ_Sh([f,g]) ≈ D(μ_Sh) + 𝔟_{μ_Sh}(Φ_ω(f),Φ_ω(g)). They show that 𝔟_{μ_Sh} controls the extendability of μ_Sh and the triviality of Reznikov’s class on subgroups, and they obtain explicit expressions for 𝔟_{μ_Sh} in several constructions such as products of surfaces and blow-ups, yielding concrete flux constraints for commuting symplectomorphisms. The results provide a closed-manifold analogue of Rousseau’s open-manifold pairing and offer a versatile framework to study flux, extendability, and characteristic classes via invariant quasimorphisms, with further implications for Calabi-type quasimorphisms on surfaces and Reznikov theory. Overall, the paper develops a robust, axiomatized approach to relate quasimorphisms to flux data and cohomological invariants in symplectic topology, with broad applicability to explicit geometric constructions.

Abstract

Given a closed connected symplectic manifold $(M,ω)$, we construct an alternating $\mathbb{R}$-bilinear form $\mathfrak{b}=\mathfrak{b}_{μ_{\mathrm{Sh}}}$ on the real first cohomology of $M$ from Shelukhin's quasimorphism $μ_{\mathrm{Sh}}$. Here $μ_{\mathrm{Sh}}$ is defined on the universal cover of the group of Hamiltonian diffeomorphisms on $(M,ω)$. This bilinear form is invariant under the symplectic mapping class group action, and $\mathfrak{b}$ yields a constraint on the fluxes of commuting two elements in the group of symplectomorphisms on $(M,ω)$. These results might be seen as an analog of Rousseau's result for an open connected symplectic manifold, where he recovered the symplectic pairing from the Calabi homomorphism. Furthermore, $\mathfrak{b}$ controls the extendability of Shelukhin's quasimorphisms, as well as the triviality of a characteristic class of Reznikov. To construct $\mathfrak{b}$, we build general machinery for a group $G$ of producing a real-valued $\mathbb{Z}$-bilinear form $\mathfrak{b}_μ$ from a $G$-invariant quasimorphism $μ$ on the commutator subgroup of $G$.

Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism

TL;DR

The paper develops a general method to extract an invariant ℝ-valued bilinear form from a Symp^c(M,ω)–invariant homogeneous quasimorphism μ on the commutator subgroup N=[G,G], constructing 𝔟_μ on the abelianization Γ=G/N via a limit formula. Specializing to G=̃Symp^c(M,ω) and N=̃Ham^c(M,ω) with μ=μ_Sh, the authors prove the existence of an 𝔟_{μ_Sh} that is Symp^c(M,ω)–invariant and satisfies μ_Sh([f,g]) ≈ D(μ_Sh) + 𝔟_{μ_Sh}(Φ_ω(f),Φ_ω(g)). They show that 𝔟_{μ_Sh} controls the extendability of μ_Sh and the triviality of Reznikov’s class on subgroups, and they obtain explicit expressions for 𝔟_{μ_Sh} in several constructions such as products of surfaces and blow-ups, yielding concrete flux constraints for commuting symplectomorphisms. The results provide a closed-manifold analogue of Rousseau’s open-manifold pairing and offer a versatile framework to study flux, extendability, and characteristic classes via invariant quasimorphisms, with further implications for Calabi-type quasimorphisms on surfaces and Reznikov theory. Overall, the paper develops a robust, axiomatized approach to relate quasimorphisms to flux data and cohomological invariants in symplectic topology, with broad applicability to explicit geometric constructions.

Abstract

Given a closed connected symplectic manifold , we construct an alternating -bilinear form on the real first cohomology of from Shelukhin's quasimorphism . Here is defined on the universal cover of the group of Hamiltonian diffeomorphisms on . This bilinear form is invariant under the symplectic mapping class group action, and yields a constraint on the fluxes of commuting two elements in the group of symplectomorphisms on . These results might be seen as an analog of Rousseau's result for an open connected symplectic manifold, where he recovered the symplectic pairing from the Calabi homomorphism. Furthermore, controls the extendability of Shelukhin's quasimorphisms, as well as the triviality of a characteristic class of Reznikov. To construct , we build general machinery for a group of producing a real-valued -bilinear form from a -invariant quasimorphism on the commutator subgroup of .

Paper Structure

This paper contains 43 sections, 70 theorems, 139 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation of our results: constructing bilinear forms out of invariant quasimorphisms
  3. Main result 1: constructing an $\mathbb{R}$-bilinear form $\mathfrak{b}_{\mu_{\rm Sh}}$ from Shelukhin's quasimorphism $\mu_{\rm Sh}$
  4. Main result 2: $\mathfrak{b}_{\mu_{\rm Sh}}$ controls the extendability of $\mu_{\rm Sh}$
  5. Main result 3: $\mathfrak{b}_{\mu_{\rm Sh}}$ controls the triviality of the Reznikov class
  6. Main result 4: the explicit expression of $\mathfrak{b}_{\mu_{\rm Sh}}$
  7. Strategy of the proofs of Theorem \ref{['mthm=RRbilinear']} and Theorem \ref{['mthm=explicitShelukhin']}
  8. Notation and conventions
  9. Applications of the $\mathbb{R}$-bilinear form $\mathfrak{b}_{\mu_{\rm Sh}}$ and organization of the present paper
  10. Applications of the $\mathbb{R}$-bilinearity of $\mathfrak{b}_{\mu_{\rm Sh}}$
  11. Applications of the $\mathbb{R}$-bilinearity in the setting of other invariant quasimorphisms
  12. Applications to the flux homomorphism
  13. Applications of the explicit expression of $\mathfrak{b}_{\mu_{\rm Sh}}$
  14. Organization of the present paper
  15. Preliminaries
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $(M,\omega)$ be a $2n$-dimensional open connected symplectic manifold. Then, for all $f,g\in \widetilde{\mathrm{Symp}_0^c}(M,\omega)$, we have

Figures (2)

  • Figure 1: Vector field defining $\sigma_{\mathscr{A}}$, $\tau_{\mathscr{A}}$, $\sigma'_{\mathscr{A}}$ and $\tau'_{\mathscr{A}}$
  • Figure 2: Curves $\alpha_1,\beta_1, \alpha_2, \beta_2, \ldots,\alpha_l,\beta_l$ on $S$

Theorems & Definitions (139)

  • Theorem 1.1: Rousseau Rou
  • Theorem 1.2: constructing $\mathfrak{b}_{\omega}$ out of $\mu_{\rm Py}$
  • Remark 1.3
  • Corollary 1.4: KKMM2
  • Theorem A: constructing $\mathfrak{b}_{\mu_{\rm Sh}}$ out of $\mu_{\rm Sh}$
  • Remark 1.5
  • Theorem B: extendability of $\mu_{\rm Sh}$
  • Theorem C: characterization of the triviality of the Reznikov class
  • Theorem D: explicit expressions of $\mathfrak{b}_{\mu_{\rm Sh}}$
  • Theorem 2.1
  • ...and 129 more