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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


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.


operad, little cubes, embedding, spheres, diffeomorphism

Mathematical Subject Classification

Primary: 57R40

Secondary: 55Q45, 57M25, 57R50


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

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