On the rational homology of high dimensional analogues of spaces of long knots

Abstract

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of ${\mathbb{R}}^m$ into ${\mathbb{R}}^n$. We view the space of embeddings as the value of a certain functor at ${\mathbb{R}}^m$, and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little disks operad. We then show that the formality of the little disks operad has implications for the homological behavior of the Taylor tower. Our second result says that when $2m+1<n$, the singular chain complex of these spaces of embeddings is rationally equivalent to a direct sum of certain finite chain complexes, which we describe rather explicitly.

This is a substantial rewrite of the previous version, incorporating suggestions of two referees. We simplified the description of the category representing infinitesimal bimodules (called "weak bimodules" in the previous version). We also eliminated all mentions of discretized operads, and our results are now formulated in terms of modules over the standard little disks operad

References in corpus (1)

• Context-free manifold calculus and the Fulton-MacPherson Operad

Cited by in corpus (16)

• Spaces of smooth embeddings and configuration categories
• On the formality of the little disks operad in positive characteristic
• Swiss-cheese action on the totalization of operads under the monoid actions actions operad
• Commutative hairy graphs and representations of $Out(F_r)$
• The rational homology of spaces of long links
• Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
• Non-formality of the odd dimensional framed little balls operads
• Models for knot spaces and Atiyah duality
• Embedding calculus and grope cobordism of knots
• Quillen cohomology of enriched operads
• Low stages of the Taylor tower for r-immersions
• A Weiss-Williams theorem for spaces of embeddings and the homotopy type of spaces of long knots
• Hochschild and cotangent complexes of operadic algebras
• Cocycles of the space of long embeddings and BCR graphs with more than one loop
• Sinha's spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad
• A Kuenneth theorem for configuration spaces