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
