Volume 5, issue 4 (2005)

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The space of intervals in a Euclidean space

Shingo Okuyama

Algebraic & Geometric Topology 5 (2005) 1555–1572
 arXiv: math.AT/0511645
Abstract

For a path-connected space $X$, a well-known theorem of Segal, May and Milgram asserts that the configuration space of finite points in ${ℝ}^{n}$ with labels in $X$ is weakly homotopy equivalent to ${\Omega }^{n}{\Sigma }^{n}X$. In this paper, we introduce a space ${\mathsc{ℐ}}_{n}\left(X\right)$ of intervals suitably topologized in ${ℝ}^{n}$ with labels in a space $X$ and show that it is weakly homotopy equivalent to ${\Omega }^{n}{\Sigma }^{n}X$ without the assumption on path-connectivity.

Keywords
configuration space, partial abelian monoid, iterated loop space, space of intervals
Primary: 55P35
Secondary: 55P40