Tara Brendle, Allen Hatcher
Published 2008-05-28, updated 2010-01-11Version 2
The main result in this paper is that the space of all smooth links in Euclidean 3-space isotopic to the trivial link of n components has the same homotopy type as its finite-dimensional subspace consisting of configurations of n unlinked Euclidean circles (the "rings" in the title). There is also an analogous result for spaces of arcs in upper half-space, with circles replaced by semicircles (the "wickets" in the title). A key part of the proofs is a procedure for greatly reducing the complexity of tangled configurations of rings and wickets. This leads to simple methods for computing presentations for the fundamental groups of these spaces of rings and wickets as well as various interesting subspaces. The wicket spaces are also shown to be K(G,1)'s.
Comments: 28 pages. Some revisions in the exposition
The Groups $G_{n}^{k}$ and Fundamental Groups of Configuration Spaces
Quantum deformations of fundamental groups of oriented 3-manifolds
Constructions of surface bundles with rank two fundamental groups
![QR code for arXiv:0805.4354 [math.GT]](http://oss.arxitics.com/images/favicon.svg)