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

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.

1-loop graphs and configuration space integral for embedding spaces

“…The nontriviality results of this paper might be generalized for graphs with one or more loop components, if the corresponding forms were proved to be closed. There might be other generalizations as in [Wa2]. Indeed, some cocycles of Emb (R k , R 2k+1 ) are constructed by a method which can be considered as a generalization of the construction of this paper.…”

Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{Diff}(S^4)$ (homological interpretation)

“…This is compatible with a result of Corollary 7.15]) that there is at least one dimensional nontrivial subspace in the (+1)-eigenspace of π i (BDiff(D d , ∂))⊗Q for some i in 2d−9 ≤ i ≤ 2d−5 (the fourth band), d ≥ 6 even, as pointed out in [KRW]. As also pointed out in [KRW,Example 6.9] The method of this paper is essentially the same as [Wa2], where we studied the rational homotopy groups of Diff(D 4k−1 , ∂). Namely, we construct some explicit fiber bundles from trivalent graphs, by giving a higher-dimensional analogue of graph-clasper surgery, developed by Goussarov and Habiro for knots and 3manifolds ( [Gou, Hab]).…”

“…However, in that proof it is unavoidable to describe thorough detailed arguments of transversality and orientation, which makes the paper surprisingly long, due to the inefficiency of the author. In this paper we attempted to make the proof accessible to more readers and gave a proof of Theorem 1.1 by means of differential forms (or algebraic topology), as in [Wa2]. It is easier for the author to write shorter proof with differential forms, though the main body of the proof is compressed into one lemma, whose proof is abstract and long.…”

Bott–Taubes–Vassiliev cohomology classes by cut-and-paste topology

“…Parametrizing these integrals by the space of knots, one can view them as integrals along the fiber of a bundle over that space. Configuration space integrals produce all finite-type knot invariants [34,39], as well as real cohomology classes in spaces of knots [7] and links [17], invariants of homology 3-spheres [19,21], and characteristic classes of homology sphere bundles [40,41]. All of these ideas were outlined in the visionary work of Kontsevich [12,13].…”

“…We suspect that our construction is adaptable to the setting of characteristic classes of bundles of homology spheres [40,41,19,21], roughly generalizing from S d to homology spheres. We leave this generalization for potential future work.…”