Configuration spaces of rings and wickets

Abstract

The main result in this paper is that the space of all smooth links in ${\mathbb{R}}^3$ 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 aspherical.