Mathematics Colloquia and Seminars

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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. It can be thought of as a union of configuration spaces of distinct unordered points in X, or as the quotient of the symmetric product obtained by forgetting multiplicities. We'll show that the kth finite subset space of a connected cell complex is (k-2)-connected. This complements a result due to David 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.