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The space of intervals in a Euclidean space. (English) Zbl 1084.55008
G. Segal [Invent. Math. 21, 213–221 (1973; Zbl 0267.55020)] introduced a configuration space \(C(\mathbb R^n,X)\) of a finite number of points in \(\mathbb R^n\) with labels in \(X\), and he showed that it is weakly homotopy equivalent to \(\Omega^n\Sigma^nX\) if \(X\) is connected. When \(X\) is not connected, \(\Omega^n\Sigma^nX\) is a group completion of \(C(\mathbb R^n;X)\).
In this paper, the author defines a space \(I_n(X)\) of intervals suitably topologized in \(\mathbb R^n\) and he shows that it is weakly homotopy equivalent \(\Omega^n\Sigma^nX\) without the assumption on connectivity. For proving this result, he uses the notions of the configuration space with labels in a partial abelian monoid and several properties of it.
J. Carso and S. Warner, [Trans. Am. Math. Soc. 265, 147–162 (1981; Zbl 0475.55003)] also constructed another such group completion model based on the space of little cubes but it seems that the present one is slightly simpler.
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55P40 Suspensions
55P48 Loop space machines and operads in algebraic topology
55P35 Loop spaces
Full Text: DOI EuDML arXiv
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