Connectivity of finite subset spaces of cell complexes

Abstract

The k-th finite subset space of a topological space X is the space exp k X of nonempty subsets of X of size at most k, topologised as a quotient of X k . Using results from our earlier paper on the finite subset spaces of connected graphs we show that the k-th finite subset space of a connected cell complex is (k - 2)-connected, and (k - 1)-connected if in addition the underlying space is simply connected. We expect exp k X to be (k + m - 2)-connected if X is an m-connected cell complex, and reduce proving this to the problem of proving it for finite wedges of (m+1)-spheres. Our results complement a theorem due to Handel that for path-connected Hausdorff X the map on π i induced by the inclusion exp k X → exp 2k+1 X is zero for all k ≥ 1 and i ≥ 0.