The space of intervals in a Euclidean space

Okuyama, Shingo

doi:10.2140/agt.2005.5.1555

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The space of intervals in a Euclidean space

Shingo Okuyama

Algebraic & Geometric Topology 5 (2005) 1555–1572

DOI: 10.2140/agt.2005.5.1555

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 $Ω^{n} Σ^{n} X$ . In this paper, we introduce a space $ℐ_{n} (X)$ of intervals suitably topologized in $ℝ^{n}$ with labels in a space $X$ and show that it is weakly homotopy equivalent to $Ω^{n} Σ^{n} X$ without the assumption on path-connectivity.

Keywords

configuration space, partial abelian monoid, iterated loop space, space of intervals

Mathematical Subject Classification 2000

Primary: 55P35

Secondary: 55P40

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Publication

Received: 15 December 2003

Revised: 25 March 2005

Accepted: 10 November 2005

Published: 23 November 2005

Authors

Shingo Okuyama
Takuma National College of Technology Kagawa 769-1192 Japan

Volume 5, issue 4 (2005)