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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.
MSC:
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55P40 Suspensions
55P48 Loop space machines and operads in algebraic topology
55P35 Loop spaces
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References:
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