Labeled configuration spaces and group completions

Abstract

Given a pair of a partial abelian monoid M and a pointed space X, let CM(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 CM(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, CM(ℝ∞, X) is weakly equivalent to the infinite loop space Ω∞Σ∞X if X is connected.