Deloopings of the spaces of long embeddings

Keiichi Sakai¹

¹ Department of Mathematical Sciences Shinshu University 3-1-1 Asahi, Matsumoto Nagano 390-8621, Japan

Fundamenta Mathematicae, Tome 227 (2014) no. 1, pp. 27-34.

Résumé

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.

Détail

DOI : 10.4064/fm227-1-3

Keywords: homotopy fiber inclusion long embedding space long immersion space known iterated based loop space codimension greater paper deloop homotopy fiber obtain topological stiefel manifold combining results lashof lees delooping long embedding space which regarded version morlet burghelea lashofs delooping diffeomorphism group disk relative boundary corollary homotopy fiber weakly equivalent space which framed little disks operad acts possibly nontrivially hence its rational homology higher bv algebra stable range dimensions