Connectivity and homology of finite subset spaces of cardinality at most four
For a space \(X\), let \(\text{Sub}_n X\) denote the space of all subsets of \(X\) of cardinality at most \(n\), with the quotient topology of \(X^n\) by the identification \((x_1, \dots , x_n) \sim (y_1, \dots, y_n)\) if and only if \(\{x_1 , \dots , x_n \} = \{y_1, \dots, y_n\}\). If \(X\) is a Hausdorff space, then the topology of \(\text{Sub}_n X\) is equal to the Vietoris topology. \textit{C. Tuffley} [Pac.\ J.\ Math.\ 217, No.~1, 175--179 (2004; Zbl 1064.55014)] proved that if \(X\) is path-connected, then \(\text{Sub}_n X\) is \((n-2)\)-connected, and he conjectured that if \(X\) is \(r\)-connected, then \(\text{Sub}_n X\) is \((n+r-2)\)-connected. \textit{S. Kallel} and \textit{D. Sjerve} [Homology Homotopy Appl.\ 11, No.~2, 229--250 (2009; Zbl 1186.55010)] proved that Tuffley's conjecture holds for \(n=3\). In the paper under review, the author answers the conjecture in the case of \(n=4\) by proving that if \(X\) is \(r\)-connected, \(r>1\), then \(\text{Sub}_4 X\) is \((r+2)\)-connected. This is proved by showing that the \((r+2)\)nd homology of \(\text{Sub}_4 X\) vanishes. The author also computes the Euler characteristic of \(\text{Sub}_4 X\) and determines the rational homotopy type of \(\text{Sub}_4 S^d\) for the \(d\)-dimensional sphere \(S^d\).
scientific article; zbMATH DE number 5935914
scientific article