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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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