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 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}}.

Commutative hairy graphs and representations of Out (Fr)

“…The complexes HGC r,h,I m,n and HGC r,h,II m,n are defined as the direct summands of HGC r,h m,n arising through this splitting into these two coefficient systems and isomorphism (20). Since these coefficient systems are formal, one has…”

“…The construction of the spectral sequences above can be extended to more general 'hairy' graph complexes considered in the literature. For example, it is shown in [20] that under suitable hypothesis the rational homotopy of the space of long embeddings (modulo immersions) of N 'strings' of dimensions m 1 , . .…”

Hochschild-Pirashvili homology on suspensions and representations of Out(F_n)

“…Indeed, the most surprising and perhaps the motivating result for T. Pirashvili to write his seminal work [28] was the striking fact that the higher Hochschild homology on a sphere of any positive dimension also admits the Hodge splitting and moreover the terms of the splitting up to a regrading depend only on the parity of the dimension of the sphere. With this excuse to be born, the higher Hochschild homology is nowadays a widely used tool that has various applications including the string topology and more generally the study of mapping and embedding spaces [28,1,2,15,25,26,30,31]. It also has very interesting and deep generalizations such as the topological higher Hochschild homology [8,29] and factorization homology [3,14,16,23].…”

“…acknowledges partial support by the Swiss National Science Foundation (grant 200021 150012) and the SwissMap NCCR, funded by the Swiss National Science Foundation. 1 These representations appear as application to the hairy graph-homology computations in the study of the spaces of long embeddings, higher dimensional string links, and the deformation theory of the little discs operads [2,31,32,33,34].…”

“…The higher Hochschild homology on spheres was introduced and studied in the original work of Pirashvili [28] and on wedges of spheres it was studied in [30,31] in connection with the homology and homotopy of spaces of higher dimensional string links. An interesting feature of this homology is that it admits a decomposition into a direct product, and the factors of this Hodge splitting depend only on the parity of the dimensions of the spheres.…”