Algebraic and Geometric Topology 5 (2005), paper no. 62, pages 1555-1572.

The space of intervals in a Euclidean space

Shingo Okuyama

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 the n-th loop-suspension of 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 n-th loop-suspension of X without the assumption on path-connectivity.

Keywords. Configuration space, partial abelian monoid, iterated loop space, space of intervals

AMS subject classification. Primary: 55P35. Secondary: 55P40.

E-print: arXiv:math.AT/0511645

DOI: 10.2140/agt.2005.5.1555

Submitted: 15 December 2003. (Revised: 25 March 2005.) Accepted: 10 November 2005. Published: 23 November 2005.

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Shingo Okuyama
Takuma National College of Technology
Kagawa 769-1192, JAPAN

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