Mathematics – Algebraic Topology
Scientific paper
2006-05-02
Geom. Topol. Monogr. 13 (2008) 41-83
Mathematics
Algebraic Topology
This is the version published by Geometry & Topology Monographs on 22 February 2008
Scientific paper
10.2140/gtm.2008.13.41
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.
No associations
LandOfFree
A family of embedding spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A family of embedding spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A family of embedding spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-184491