On the Other Side of the Bialgebra of Chord Diagrams

Abstract

In this paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots inThe first term of the Vassiliev spectral sequence is concentrated in some angle of the second quadrant. In homological case the lower line of this term is the bialgebra of chord diagrams (or its superanalog if d is even). We prove in this paper that the groups of the upper line are all trivial. In the same bigradings we compute the homology groups of the complex spanned only by strata of immersions in the discriminant (maps having only self-intersections). We interprete the obtained groups as subgroups of the (co)homology groups of the double loop space of a (d − 1)-dimensional sphere. In homological case the last complex is the normalized Hochschild complex of the Poisson or Gerstenhaber (depending on parity of d) algebras operad. The upper line bigradings are spanned by the operad of Lie algebras. To describe the cycles in these bigradings we introduce new homological operations on Hochschild complexes. These new operations are in fact the Dyer-Lashof operations induced by the action of the singular chains operad of little squares on Hochschild complexes.

Dyer–Lashof–Cohen operations in Hochschild cohomology

“…Proof of Theorem 3.1 We proved in [25,Section 11], that if x is an odd degree cycle in a Hochschild complex (or of any degree with characteristic p D 2), then the following formula holds: 2…”

Lie coalgebras and rational homotopy theory, I: graph coalgebras

“…There is a similar set of relations which the configuration pairing respects among graphs. The first author's paper [19] establishes the following theorem, which was first proven independently by Tourtchine [24] and, in the odd setting, Melancon and Reutenauer [11]. Theorem 2.16.…”

Poisson structures on the homology of the space of knots

“…When moreover the cosimplicial space arises from the operad O with multiplication, then its E 1 -term is the Hochschild complex of the homology operad H * (O). It is known (Gerstenhaber-Voronov [6], Turchin [17,18]) that there exist a natural product and a bracket on such a complex which induce the Gerstenhaber algebra structure, the degree one Poisson algebra structure, on the homology. Note that the Gerstenhaber algebra structure also comes from the action of the chains of the little disks operad (Deligne's conjecture; see McClure-Smith [13]).…”

“…In some cases this gives us a method to compute the (topological) Browder operation purely algebraically, as was done by Turchin [18].…”

“…We regard λ as a Poisson algebra structure on the homology (see Cohen [5] • is an operad with multiplication, there exist a natural product and a bracket on E 1 -term (see Gerstenhaber-Voronov [6], Turchin [17,18]) that make the E 2 -term HH * (O • ) a Gerstenhaber algebra, that is, a Poisson algebra of degree one [17,18]. This Poisson structure defined on E 2 -term in fact descends on further terms (Proposition 4.8).…”