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Labeled configuration spaces and group completions

Labeled configuration spaces and group completions Given a pair of a partial abelian monoid M and a pointed space X , let C M (ℝ ∞ , X ) denote the configuration space of finite distinct points in ℝ ∞ parametrized by the partial monoid X ∧ M . In this note we will show that if M is embedded in a topological abelian group and if we put ± M = { a − b | a , b ∈ M } then the natural map C M (ℝ ∞ , X ) → C ± M (ℝ ∞ , X ) induced by the inclusion M ⊂ ± M is a group completion. This result can be applied to show that for any finite set M such that {0} ⊊ M ⊂ ℤ, C M (ℝ ∞ , X ) is weakly equivalent to the infinite loop space Ω ∞ Σ ∞ X if X is connected. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Labeled configuration spaces and group completions

Shimakawa, Kazuhisa
Forum Mathematicum , Volume 19 (2) – Mar 20, 2007
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References (7)

Publisher
de Gruyter
Copyright
© Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2007.014
Publisher site
See Article on Publisher Site

Abstract

Given a pair of a partial abelian monoid M and a pointed space X , let C M (ℝ ∞ , X ) denote the configuration space of finite distinct points in ℝ ∞ parametrized by the partial monoid X ∧ M . In this note we will show that if M is embedded in a topological abelian group and if we put ± M = { a − b | a , b ∈ M } then the natural map C M (ℝ ∞ , X ) → C ± M (ℝ ∞ , X ) induced by the inclusion M ⊂ ± M is a group completion. This result can be applied to show that for any finite set M such that {0} ⊊ M ⊂ ℤ, C M (ℝ ∞ , X ) is weakly equivalent to the infinite loop space Ω ∞ Σ ∞ X if X is connected.

Journal

Forum Mathematicum – de Gruyter

Published: Mar 20, 2007

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