Connectivity of finite subset spaces of cell complexes
arXiv:math/0304086 · doi:10.2140/pjm.2004.217.175
Abstract
The kth finite subset space of a topological space X is the space exp_k X of non-empty subsets of X of size at most k, topologised as a quotient of X^k. Using results from our earlier paper (math.GT/0210315) on the finite subset spaces of connected graphs we show that the kth 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 pi_i induced by the inclusion exp_k X --> exp_{2k+1} X is zero for all k and i.
4 pages. v2: retitled; main result strengthened for simply-connected complexes using math.GT/0311371, and problem of strengthening it for m-connected complexes reduced to strengthening it for wedges of (m+1)-spheres