  Let Emb(S^j,S^n) denote the space of C^infty-smooth embeddings of the
j-sphere in the n-sphere. This paper considers homotopy-theoretic properties of
the family of spaces Emb(S^j,S^n) for n >= j > 0. There is a
homotopy-equivalence of Emb(S^j,S^n) with SO_{n+1} times_{SO_{n-j}} K_{n,j}
where K_{n,j} is the space of embeddings of R^j in R^n which are standard
outside of a ball. The main results of this paper are that K_{n,j} is
(2n-3j-4)-connected, the computation of pi_{2n-3j-3} (K_{n,j}) together with a
geometric interpretation of the generators. A graphing construction Omega
K_{n-1,j-1} --> K_{n,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 pi_0
(Emb(S^j,S^n)) 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 K_{n,j},
focusing on issues related to iterated loop-space structures.
