Deformation quantization and the Gerstenhaber structure on the homology of knot spaces

Abstract

It is well known that (in suitable codimension) the spaces of long knots in R n modulo immersions are double loop spaces. Hence the homology carries a natural Gerstenhaber structure, given by the Gerstenhaber structure on the Hochschild homology of the n-Poisson operad. In this paper, we compute the latter Gerstenhaber structure in terms of hairy graphs, and show that it is not quite trivial combinatorially. Curiously, the construction makes essential use of methods and results of deformation quantization, and thus provides a bridge between two previously little related subjects in mathematics.

Little discs operads, graph complexes and Grothendieck-Teichmüller groups

“…In the case m = 1, we have to consider a deformation of this Lie dgalgebra structure which we call the Shoikhet L ∞ -structure. We just refer to [61] for the explicit definition of this structure. (Recall simply that an L ∞ -algebra denotes the structure of a strongly homotopy Lie algebra.…”

Pre-Lie pairs and triviality of the Lie bracket on the twisted hairy graph complexes

“…Recollection from [22]. We recall from [22] that the hairy graph complexes HGC 1,n are naturally equipped with a nontrivial L ∞ -structure, called in loc. cit.…”

“…where F r ∈ C(FM 1 (r)) ⊂ C(FM 2 (r)) is the fundamental chain of the connected component of FM 1 (r) in which the r points are in ascending order on the line. The link to the hairy graph complexes is that (as shown in [22]) there is an L ∞ -quasi-morphism (31) U Sh : fHGC ′ 1,2 → Conv(Assoc ∨ , Graphs 2 ) from the full (disconnected) hairy graph complex fHGC 1,2 equipped with the Shoikhet L ∞ struture. By the Goldman-Millson Theorem [5] we can hence transfer our Maurer-Cartan element to a Maurer-Cartan element in fHGC ′ 1,2 .…”

The Cohomology of the Full Directed Graph Complex

“…Graph complexes provide us with a large supply of intriguing questions and conjectures 1 [3], [5], [6], [8], [9], [13], [14], [16], [17], [19], [20], [26], [27], [28], [29]. One source of the motivation for working with graph complexes comes from the study of embedding spaces [2], [4], [21], [25], [26]. Another source [12], [15], [24] comes from the study of moduli spaces of smooth complex algebraic curves.…”