Volume 13 (2008)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
MSP Books and Monographs
About this Series
Editorial Board
Ethics Statement
Author Index
Submission Guidelines
Author Copyright Form
Purchases
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
Other MSP Publications

A family of embedding spaces

Ryan Budney

Geometry & Topology Monographs 13 (2008) 41–83

DOI: 10.2140/gtm.2008.13.41

arXiv: math.AT/0605069

Abstract

Let Emb(Sj,Sn) denote the space of C–smooth embeddings of the j–sphere in the n–sphere. This paper considers homotopy-theoretic properties of the family of spaces Emb(Sj,Sn) for n ≥ j> 0. There is a homotopy-equivalence of Emb(Sj,Sn) with SOn+1 ×SOn-j Kn,j where Kn,j is the space of embeddings of Rj in Rn which are standard outside of a ball. The main results of this paper are that Kn,j is (2n-3j-4)–connected, the computation of π2n-3j-3 Kn,j together with a geometric interpretation of the generators. A graphing construction Ω Kn-1,j-1 → Kn,j is shown to induce an epimorphism on homotopy groups up to dimension 2n-2j-5. This gives a new proof of Haefliger's theorem that π0 Emb(Sj,Sn) is a group for n-j>2. The proof given is analogous to the proof that the braid group has inverses. Relationship between the graphing construction and actions of operads of cubes on embedding spaces are developed. The paper ends with a brief survey of what is known about the spaces Kn,j, focusing on issues related to iterated loop-space structures.

Keywords

operad, little cubes, embedding, spheres, diffeomorphism

Mathematical Subject Classification

Primary: 57R40

Secondary: 55Q45, 57M25, 57R50

References
Publication

Received: 7 May 2006
Revised: 24 June 2007
Accepted: 2 July 2007
Published: 22 February 2008

Authors
Ryan Budney
Mathematics and Statistics
University of Victoria
PO Box 3045 STN CSC
Victoria
British Columbia
V8W 3P4
Canada