#### Volume 7, issue 4 (2007)

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The homotopy infinite symmetric product represents stable homotopy

### Christian Schlichtkrull

Algebraic & Geometric Topology 7 (2007) 1963–1977
##### Abstract

We modify the definition of the infinite symmetric product of a based space $X$ by applying the homotopy colimit instead of the colimit. This gives a topological monoid $S{P}_{h}\left(X\right)$ and using formal properties of homotopy colimits, we prove that its group completion represents the stable homotopy of $X$. In this way we get a streamlined approach to the Barratt–Priddy–Quillen theorem.

##### Keywords
homotopy colimit, infinite symmetric product, Barratt–Priddy–Quillen Theorem
Primary: 55Q10
Secondary: 55P42
##### Publication
Received: 10 April 2006
Accepted: 5 November 2007
Published: 18 December 2007
##### Authors
 Christian Schlichtkrull Department of Mathematics University of Bergen Johannes Brunsgate 12 5008 Bergen Norway