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Configurations and parallelograms associated to centers of mass. (English) Zbl 1170.55010
Hubbuck, John (ed.) et al., Proceedings of the school and conference in algebraic topology, the Vietnam National University, Hanoi, Vietnam, August 9–20, 2004. In honor of Huỳnh Mùi on the occasion of his 60th birthday. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 11, 17-32 (2007).
Let \(k\) and \(t\) be two integers such that \(1 \leq t \leq k\). The authors define the subspace \(M(t,k)\) of \(\mathbb C^k\) of \(k\)-tuples \((z_1,\dots,z_k)\) of distinct points such that the centroids of the subsets of \(\{z_1,\dots,z_k\}\) with \(t\) elements are distinct. Let \(\text{Conf}(\mathbb C,k)\) be the configuration space of ordered \(k\)-tuples of distinct points in \(\mathbb C\) and let \( M(t,k) \rightarrow \text{Conf}(\mathbb C,k)\) denote the \({\mathfrak S}_k\)-natural inclusion map. The authors show that the induced map \( M(2,k)/{\mathfrak S}_k\rightarrow \text{Conf}(\mathbb C,k)/{\mathfrak S}_k\) is not a surjection in \(\operatorname {mod} 2\) homology and thus it does not admit a cross-section up to homotopy.
The authors point out that their methods fail to generalize to the maps \(M(p,k)/{\mathfrak S}_k\rightarrow \text{Conf}({\mathcal C},k)/{\mathfrak S}_k\) for \(p\) an odd prime. This leads to a speculation concerning the localization of the double loop space of a sphere at an odd prime.
For the entire collection see [Zbl 1133.57003].