Rational homology and homotopy of high dimensional string links

TL;DR

This paper extends the calculation of rational homology and homotopy from high dimensional knots to string links, using graph complexes and Hochschild homology, with conjectures on broader applicability.

Contribution

It generalizes existing results to high dimensional string links and proposes a conjecture that the homotopy graph-complex computes rational homotopy groups in all codimensions greater than two.

Findings

Confirmed the conjecture at the level of a_0 using Haefliger's approach.

Provided explicit graph-complex models for rational homology and homotopy of high dimensional string links.

Extended the applicability range of these models to higher codimensions.

Abstract

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high dimensional anologues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher order Hochschild homology also called Hochschild-Pirashvili homology. In this paper, we generalize all these results to high dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under the study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of links spaces always when codimension is greater than two, i.e. always when the Goodwillie-Weiss calculus is applicable. Using Haefliger\rq{}s approach to calculate the groups of isotopy classes of higher dimensional links, we confirm our cojecture at the level of \pi_0.