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Characteristic classes of framed fibre bundles

Nils Prigge

TL;DR

This work extends Kontsevich’s graph-based characteristic classes from disk-like fibres to framed fibre bundles with closed fibres by developing a comprehensive algebraic and geometric framework around Fulton–MacPherson configuration spaces, twisted graph complexes, and complete dg Lie algebras. It constructs a fibrewise partition function Z_E and an I-map from a Chevalley–Eilenberg complex to the base's de Rham complex, yielding characteristic classes that are natural and pullback-compatible for framed M-bundles with trivial base action on cohomology. The approach connects to the rational homotopy theory of E_d-modules, interprets the results via automorphism spaces of configuration spaces, and provides a structural path toward classifying maps into automorphism spaces, potentially uncovering new real homotopy invariants for families of configuration spaces. Overall, the paper links Kontsevich-type invariants to embedding calculus and the broader theory of FM_d-modules, enriching both the geometric and algebraic perspectives on fibre-bundle characteristic classes.

Abstract

We generalize Kontsevich's construction of characteristic classes of fibre bundles with homology sphere fibres and a trivialization of the vertical tangent bundle to framed fibre bundles with closed manifold fibres.

Characteristic classes of framed fibre bundles

TL;DR

This work extends Kontsevich’s graph-based characteristic classes from disk-like fibres to framed fibre bundles with closed fibres by developing a comprehensive algebraic and geometric framework around Fulton–MacPherson configuration spaces, twisted graph complexes, and complete dg Lie algebras. It constructs a fibrewise partition function Z_E and an I-map from a Chevalley–Eilenberg complex to the base's de Rham complex, yielding characteristic classes that are natural and pullback-compatible for framed M-bundles with trivial base action on cohomology. The approach connects to the rational homotopy theory of E_d-modules, interprets the results via automorphism spaces of configuration spaces, and provides a structural path toward classifying maps into automorphism spaces, potentially uncovering new real homotopy invariants for families of configuration spaces. Overall, the paper links Kontsevich-type invariants to embedding calculus and the broader theory of FM_d-modules, enriching both the geometric and algebraic perspectives on fibre-bundle characteristic classes.

Abstract

We generalize Kontsevich's construction of characteristic classes of fibre bundles with homology sphere fibres and a trivialization of the vertical tangent bundle to framed fibre bundles with closed manifold fibres.
Paper Structure (13 sections, 25 theorems, 73 equations)

This paper contains 13 sections, 25 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compactifications of configuration spaces
  4. Real models for configuration spaces
  5. Complete dg Lie algebras
  6. Models for fibrations
  7. Propagators for bundles
  8. Graph characteristic classes
  9. Connection to automorphisms of configuration spaces
  10. Cooperadic twisting
  11. The (full) graph complex via operadic twisting
  12. Orientation conventions and fibre integration
  13. Orientation of the boundary faces of FMM

Key Result

Theorem 1

Let $\pi:E{\rightarrow} B$ be a framed fibre bundle of connected spaces with closed fibre $M$ and $\dim M>2$ so that $\pi_1(B)$ acts trivially on $H^*(M;\mathbb{R})$. One can define a fibrewise partition function via configuration space integrals which is an invariant of the framed fibre bundle that is natural with respect to pullbacks.

Theorems & Definitions (38)

  • Theorem 1: Theorem \ref{['PartFctZE']}
  • Remark 1.1
  • Theorem 2: Theorem \ref{['MainTheorem']}
  • Remark 1.2
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: CW16
  • Lemma 2.4: CW16
  • Theorem 2.5: CW23
  • Remark 2.6
  • ...and 28 more