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On Kontsevich’s characteristic classes for higher dimensional sphere bundles I: the simplest class


This paper studies the simplest one of the sequence of characteristic classes of framed smooth fiber bundles constructed by M. Kontsevich. By introducing a correction term to the characteristic number of the Kontsevich class, we obtain an invariant of unframed sphere bundles over a sphere. The correction term is given by a multiple of Hirzebruch’s signature defect. We observe that a reduction of our invariant modulo a certain integer agrees with a multiple of Milnor’s λ′-invariant of exotic spheres. Furthermore, our invariant is non-trivial for many fiber dimensions. Hence we can detect some ‘exotic’ non-trivial subspace of π i (Diff(S d)) ⊗ \({\mathbb {Q}}\) for some pairs (i, d) which are not in Igusa’s stable range.

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Correspondence to Tadayuki Watanabe.

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Dedicated to Professor Akio Kawauchi for his 60th birthday.

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Watanabe, T. On Kontsevich’s characteristic classes for higher dimensional sphere bundles I: the simplest class. Math. Z. 262, 683 (2009).

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  • Cohomology Class
  • Homotopy Class
  • Signature Defect
  • Euler Class
  • Sphere Bundle