Deloopings of the spaces of long embeddings

TL;DR

This paper advances the understanding of the topology of long embedding spaces by delooping their homotopy fibers, linking them to topological Stiefel manifolds and operad actions, with implications for their rational homology.

Contribution

It provides a new delooping of the homotopy fiber of long embeddings, connecting it to topological Stiefel manifolds and operad actions, extending previous delooping results.

Findings

Homotopy fiber is weakly equivalent to a space with framed little disks operad action.

Rational homology of the fiber forms a higher BV-algebra in certain dimensions.

Delooping of the long embedding space generalizes Morlet-Burghelea-Lashof's delooping of the diffeomorphism group.

Abstract

The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give a delooping of the long embedding space, which can be regarded as a version of Morlet-Burghelea-Lashof's delooping of the diffeomorphism group of the disk relative to the boundary. As a corollary, we show that the homotopy fiber is weakly equivalent to a space on which the framed little disks operad acts possibly nontrivially, and hence its rational homology is a (higher) BV-algebra in a stable range of dimensions.