Kontsevich's Characteristic Classes as Topological Invariants of Configuration Space Bundles

TL;DR

This paper explores how Kontsevich's characteristic classes serve as topological invariants capable of distinguishing smooth structures on fiber bundles, linking them to configuration space topology and real blow-up constructions.

Contribution

It demonstrates that Kontsevich's classes are determined by the topology of configuration space bundles and framing data, revealing their role in differentiating smooth structures.

Findings

Kontsevich's classes distinguish smooth $S^4$-bundles topologically.

The classes depend on the topology of 2-point configuration space bundles.

Real blow-up constructions relate to smooth structure detection.

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 $\pi$ are determined by the topology of the 2-point configuration space bundle of $\pi$ and framing data.