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

Abstract

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 πi(Diff(Sd)) ⊗ \({\mathbb {Q}}\) for some pairs (i, d) which are not in Igusa’s stable range.