Deloopings of the spaces of long embeddings

Tom 227 / 2014

Keiichi Sakai Fundamenta Mathematicae 227 (2014), 27-34 MSC: Primary 58D10; Secondary 55P50, 57Q45, 57N35. DOI: 10.4064/fm227-1-3


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.


  • Keiichi SakaiDepartment of Mathematical Sciences
    Shinshu University
    3-1-1 Asahi, Matsumoto
    Nagano 390-8621, Japan

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