Rational homology and homotopy of high dimensional string links

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.