Finite subset spaces of \(S^1\) (Q1855205)

The author obtains new results on the spaces \(\exp_k(S^1)\), the space of subsets (configurations) of \(k\) or fewer points in \(S^1\). For example, he shows that the space \(\exp_k(S^1)\) has the homotopy type of an odd dimensional sphere, of dimension \(k\) or \(k-1\) according to whether \(k\) is odd or even. Another result is that the complement of \(\exp_{k-2}(S^1)\) in \(\exp_k(S^1)\) has the homotopy type of the complement of a \((k-1,k)\) torus knot.