Kontsevich’s characteristic classes as topological invariants of configuration space bundles

Chen, Xujia

doi:10.2140/agt.2026.26.349

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Kontsevich's characteristic classes as topological invariants of configuration space bundles

Xujia Chen

Algebraic & Geometric Topology 26 (2026) 349–395

DOI: 10.2140/agt.2026.26.349

Abstract

Kontsevich’s characteristic classes are invariants of framed smooth fiber bundles with homology sphere fibers. It was shown by Watanabe that they can be used to distinguish smooth $S^{4}$ -bundles that are all trivial as topological fiber bundles. In this article we show that this ability of Kontsevich’s classes is a manifestation of the following principle: the “real blow-up” construction on a smooth manifold essentially depends on its smooth structure and thus, given a smooth manifold (or smooth fiber bundle) $M$ , the topological invariants of spaces constructed from $M$ by real blow-ups could potentially differentiate smooth structures on $M$ . The main theorem says that Kontsevich’s characteristic classes of a smooth framed bundle $π$ are determined by the topology of the 2-point configuration space bundle of $π$ and framing data.

Keywords

configuration space integrals, configuration space, diffeomorphism group, Kontsevich's invariants, Kontsevich's characteristic classes

Mathematical Subject Classification

Primary: 57N16, 57R20

References

Publication

Received: 30 April 2024

Revised: 28 January 2025

Accepted: 9 February 2025

Published: 16 January 2026

Authors

Xujia Chen
Max Planck Institute for Mathematics Bonn Germany

This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2026, the article will become Open Access in early 2026. Otherwise, this article (and all 2026 articles) will be available only to paid subscribers.

Volume 26, issue 1 (2026)