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.
There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2 and by Cattaneo-Rossi for higher degrees. However, its feature is yet unknown. In this paper we restrict the study to long ribbon n-knots and characterize the Bott-Cattaneo-Rossi invariant as a finite type invariant of long ribbon n-knots introduced in [HKS]. As a consequence, we obtain a non-trivial description of the Bott-Cattaneo-Rossi invariant in terms of the Alexander polynomial.The results for higher codimension knots are also given. In those cases similar differential forms to define Bott-Cattaneo-Rossi invariant yields infinitely many cohomology classes of Emb(R n , R m ) if m, n ≥ 3 odd and m > n + 2. We observe that half of these classes are non-trivial, along a line similar to Cattaneo-CottaRamusino-Longoni [CCL].
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 nontrivial for many fiber dimensions. Hence we can detect some 'exotic' non-trivial subspace of π i (Diff(S d )) ⊗ Q for some pairs (i, d) which are not in Igusa's stable range.
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