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Configured locally smooth cohomology and $\mathbb{Q}/\mathbb{Z}$-torsion in $H_3$ of diffeomorphism groups

Takefumi Nosaka

TL;DR

Nosaka develops configured locally smooth cohomology, a framework built from well-configured tuples and geometric straightening, to produce explicit locally smooth $\mathbb{R}/\mathbb{Z}$-valued $3$-cocycles on diffeomorphism groups preserving volume, symplectic, and contact structures. Central to the method is a Dupont-type chain map from well-configured complexes, realized via affine straightening in Fréchet–Lie groups, which ties geometric forms to Gel'fand–Fuks cohomology and enables explicit cocycle construction. Applying this to $\mathrm{Diff}_v(M)$, $\mathrm{Diff}_{\omega}(M)$, and $\mathrm{Diff}_{\alpha}(M)$, and using Cheeger–Chern–Simons theory together with transfer arguments, the paper proves the third group homology contains ${\mathbb{Q}}/{\mathbb{Z}}$-torsion in several cases (notably spherical space forms and certain projective cases), while treating hyperbolic and contact settings with analogous, sometimes real-valued cocycles. The results provide computable torsion phenomena in $H^{\mathrm{gr}}_3$ for diffeomorphism groups arising in low-dimensional and symplectic topology, illustrating the power of configuration-based cohomology to detect higher-degree invariants that escape the full homogeneous complex.

Abstract

We introduce configured group cohomology, a variant of locally smooth cohomology built from well-configured tuples and geometric fillings. This framework yields explicit locally smooth $\R/\Z$-valued $3$-cocycles of Chern--Simons type on diffeomorphism groups preserving geometric structures. As an application we show that, for several such groups, the third group homology contains a subgroup isomorphic to $\Q/\Z$.

Configured locally smooth cohomology and $\mathbb{Q}/\mathbb{Z}$-torsion in $H_3$ of diffeomorphism groups

TL;DR

Nosaka develops configured locally smooth cohomology, a framework built from well-configured tuples and geometric straightening, to produce explicit locally smooth -valued -cocycles on diffeomorphism groups preserving volume, symplectic, and contact structures. Central to the method is a Dupont-type chain map from well-configured complexes, realized via affine straightening in Fréchet–Lie groups, which ties geometric forms to Gel'fand–Fuks cohomology and enables explicit cocycle construction. Applying this to , , and , and using Cheeger–Chern–Simons theory together with transfer arguments, the paper proves the third group homology contains -torsion in several cases (notably spherical space forms and certain projective cases), while treating hyperbolic and contact settings with analogous, sometimes real-valued cocycles. The results provide computable torsion phenomena in for diffeomorphism groups arising in low-dimensional and symplectic topology, illustrating the power of configuration-based cohomology to detect higher-degree invariants that escape the full homogeneous complex.

Abstract

We introduce configured group cohomology, a variant of locally smooth cohomology built from well-configured tuples and geometric fillings. This framework yields explicit locally smooth -valued -cocycles of Chern--Simons type on diffeomorphism groups preserving geometric structures. As an application we show that, for several such groups, the third group homology contains a subgroup isomorphic to .
Paper Structure (21 sections, 28 theorems, 90 equations)

This paper contains 21 sections, 28 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries; group cohomology
  3. Review of group (co)homology and (locally) smooth group cohomology
  4. Configured continuous cohomology
  5. A Dupont-type chain map from well-configured complexes
  6. A Dupont-type chain map
  7. Affine simplices in Fréchet--Lie groups
  8. Well-configured complexes from $\mathop{\mathrm{Diff}}\nolimits(M)$
  9. Relation to the Gel'fand--Fuks cohomology
  10. Applications to the smooth cohomology of $\mathop{\mathrm{Diff}}\nolimits_v(M)$
  11. Review: Cheeger--Chern--Simons cocycles
  12. A lemma on transfer
  13. Proof of Theorem \ref{['main1']}
  14. The hyperbolic case
  15. Motivation: why configuration complexes are needed
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\Gamma\subset {\rm SO}(4)$ be one of the following finite subgroups: $\{e\}$, the quaternion group $Q_8$, the prism group $D_{4n}$, the tetrahedral $T_{24}$, the octahedral $O_{48}$, or the icosahedral $I_{120}$, where $n\in\mathbb{N}$. Assume that the induced action of $\Gamma$ on $S^3$ is fre

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2: cf. Comparison theorem in Bro
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • ...and 58 more