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

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.

arXiv comment54 pages. v3 update: minor changes correcting typos and improving the exposition; added a rephrasement of the main theorem; added references. v2 update: corrected minor mistakes and a gap in Section 3.2; added Section 3.3; Section 5 is largely rewritten (v1 contains major mistakes); added 2 figures