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.

Authors