On the homology of the spaces of long knots

Keywords: discriminant of the space of knots, bialgebra of chord diagrams, Hochschild complex, operads of Poisson – Gerstenhaber – Batalin-Vilkovisky algebras. This paper is a more detailed version of [T1], where the first term of the Vassiliev spectral sequence (computing the homology of the space of long knots in R d , d ≥ 3) was described in terms of the Hochschild homology of the Poisson algebras operad for d odd, and of the Gerstenhaber algebras operad for d even. In particular, the bialgebra of chord diagrams arises as some subspace of this homology. The homology in question is the space of characteristic classes for Hochschild cohomology of Poisson (resp. Gerstenhaber) algebras considered as associative algebras. The paper begins with necessary preliminaries on operads. Also we give a simplification of the computations of the first term of the Vassiliev spectral sequence. We do not give proofs of the results. 0 Introduction First we recall some known facts on the Vassiliev spectral sequence and then proceed to explaining of the main idea of the work. Let us fix a non-trivial linear map l : R 1 ֒→ R d. We will consider the space of long knots, i. e., of injective smooth non-singular maps R 1 ֒→ R d , that coincide with the map l outside some compact set (this set is not fixed). The long knots form an open everywhere dense subset in the affine space K of all smooth maps R 1 → R d with the same behavior at infinity. The complement Σ ⊂ K of this dense subset is called the discriminant space. It consists of the maps having self-intersections or singularities. Any cohomology class γ ∈ H i (K\Σ) of the knot space can be realized as the linking coefficient with an appropriate chain in Σ of codimension i + 1 in K. Following [V5] we will assume that the space K has a very large but finite dimension ω. A partial justification of this assumption uses finite dimensional approximations of K, see [V1]. Below we indicate by quotes non-rigorous assertions using this assumption and needing a reference to [V2] for such a justification. The main tool of Vassiliev's approach to computation of the (co)homology of the knot space is the simplicial resolution σ (constructed in [V1]) of the discriminant Σ. This resolution is also called the resolved discriminant. The natural projection Π : ¯ …