On the other side of the bialgebra of chord diagrams

TL;DR

This paper explores the homological structures related to the space of long knots in R^d, revealing trivial groups in certain spectral sequence lines and connecting these to operads, Hochschild complexes, and loop space homology.

Contribution

It introduces new homological operations on Hochschild complexes and establishes their relation to Dyer-Lashof operations, advancing understanding of knot space homology.

Findings

Upper line groups are trivial in the spectral sequence.

Homology groups relate to double loop space of spheres.

New operations on Hochschild complexes are introduced.

Abstract

In the 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 in R^d, d>=3. The first term of the Vassiliev spectral sequence is concentrated in some angle of the second quadrant. In homological case the lower line is the bialgebra of chord diagrams (or its superanalog if d is even). We prove 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. It will be proven elsewhere that these operations are related to Dyer-Lashof homology operations defined for double loop spaces.