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 SPh(X) 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
Mathematical Subject Classification 2000
Primary: 55Q10
Secondary: 55P42
References
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