The k-th finite subset space of a
topological space X is the space expkX of nonempty subsets of X of size at most k,
topologised as a quotient of Xk. 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 expkX 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 πi induced by the inclusion expkX↪exp2k+1X is zero for all k≥ 1 and
i ≥ 0.
