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Journal of Topology 2009 2(3):624-660; doi:10.1112/jtopol/jtp024
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© 2009 London Mathematical Society

On Kontsevich’s characteristic classes for higher-dimensional sphere bundles II: Higher classes

Tadayuki Watanabe

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku Tokyo 153-8914 Japan tadayuki@ms.u-tokyo.ac.jp


  Abstract

This paper studies Kontsevich’s characteristic classes of smooth bundles with fibre in a ‘singularly framed’ odd-dimensional homology sphere, which are defined through his graph complex and configuration space integral. We will give a systematic construction of smooth bundles parameterized by trivalent graphs and will show that our smooth bundles are non-trivially detected by Kontsevich’s characteristic classes. It turns out that there are surprisingly many non-trivial elements of the rational homotopy groups of the diffeomorphism groups of spheres that are in some ‘non-stable’ range. In particular, the homotopy groups of the diffeomorphism groups in some ‘non-stable’ range are not finite.

Received January 13, 2009.

2000 Mathematics Subject Classification 55R40 (primary), 55R10, 58D10 (secondary)

Current address: Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo Hokkaido, 060-0810 Japan tadayuki@math.sci.hokudai.ac.jp

During the preparation of this manuscript the author was supported by JSPS Research Fellowships for Young Scientists.


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