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 quotientof 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 look at this this construction applied to cell complexes, beginning with the circle. exp_2 S^1 is easily seen to be a Moebius strip (exercise!), and exp_3 S^1 is known to be S^3, with exp_1 S^1 forming a trefoil knot inside it. These results turn out to be part of a larger pattern; exp_k S^1 has the homotopy type of an odd dimensional sphere, and (exp_k S_1)\(exp_{k-2} S^1) has the homotopy type of a (k-1,k)-torus knot complement.