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

TL;DR

This paper explores the rational homology of high-dimensional spaces of long knots, using manifold calculus and operad theory to describe their homological structure and the Taylor tower approximation.

Contribution

It connects the Taylor tower of embedding spaces to infinitesimal bimodules over the little disks operad and analyzes the implications of operad formality on homology.

Findings

The Taylor tower can be expressed via maps between bimodules over the little disks operad.

The formality of the little disks operad influences the homological properties of the embedding spaces.

For 2m+1<n, the chain complex of embedding spaces is rationally equivalent to a sum of explicit finite complexes.

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.