Poisson structures on the homology of the space of knots

Abstract

In this article we study the Poisson algebra structure on the homology of the totalization of a fibrant cosimplicial space associated with an operad with multiplication. This structure is given as the Browder operation induced by the action of little disks operad, which was found by McClure and Smith. We show that the Browder operation coincides with the Gerstenhaber bracket on the Hochschild homology, which appears as the E 2 -term of the homology spectral sequence constructed by Bousfield. In particular we consider a variant of the space of long knots in higher dimensional Euclidean space, and show that Sinha's homology spectral sequence computes the Poisson algebra structure of the homology of the space. The Browder operation produces a homology class which does not directly correspond to chord diagrams.

Nontrivalent graph cocycle and cohomology of the long knot space

“…Theorem 1.1 is an analogous result to those of [12,13], but the proof is more geometric. We prove the non-triviality of I(Γ) by evaluating it on a cycle produced by the action of little disks operad on the space K n , defined in [3].…”

“…Their approaches in some senses make use of the little disks operad and its action on K n , which induces on H * ( K n ) the Browder operation, a structure of a Poisson algebra. This Poisson structure has not been well understood, and studied in [12,13,17], and so on.…”

“…Here we explain two examples of cycles made from chord diagrams, namely, e ∈ H n−3 ( K n ) and v 2 ∈ H 2(n−3) (K n ) which we will use later. For more general treatment, see [5,9,12,17]. 2.…”

Non-Formality of the Odd Dimensional Framed Little Balls Operads

“…As P is cofibrant (and any chain operad is fibrant), by the theory of model categories, there exists a weak equivalence q : P → H. We can take a cycle g ′ ∈ P(1) such that the class [g ′ ] goes to β m by p * . The pair (i(ν 1 ), g ′ ) satisfies the condition of Definition 4.1 for P. It is clear that the pairs (ν, p(g ′ )), (q(g ′ ), qi(ν 1 )) also ω(i(ν 1 ), g ′ ) ✤ o o ✤ / / ω(qi(ν 1 ), q(g ′ )) (14) By these isomorphisms, [ω(ν, p(g ′ ))] corresponds to [ω(qi(ν 1 ), q(g ′ ))]. As we show in the above, [ω(ν, p(g ′ ))] is non-zero.…”

BV-structures on the homology of the framed long knot space

“…It is not difficult to see that the result follows from the three formulas The Bousfield spectral sequence [1] is derived from the double complex {C * (f K n ( * )), d, ∂ * }, where f K n is the framed Kontsevich operad [14] (which is cyclic and multiplicative), C * is the singular chain complex functor and d is the boundary operator for singular chains. This spectral sequence is a spectral sequence of Poisson algebras [14,12]. The map B * is defined on C * (f K n ( * )) by (4.1) and commutes with both ∂ and d since τ k−1 and s k−1 are induced by continuous maps defined on f C( * ); τ is induced by the cyclic permutation of balls, and s is the forgetting map.…”