A family of embedding spaces

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 Emb(Sj, Sn) ' 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−3Kn,j together with a geometric interpretation of the generators. A graphing construction ΩKn−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 π0Emb(S, 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.