Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues 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 study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of
link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of {\pi_{0}}.
We make calculations in graph homology which further understanding of the topology of spaces of string links, in particular, calculating the Euler characteristics of finite-dimensional summands in their homology and homotopy. In doing so, we also determine the supercharacter of the symmetric group action on the positive arity components of the modular envelope of L ∞ .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.