The homotopy infinite symmetric product represents stable homotopy

Schlichtkrull, Christian

doi:10.2140/agt.2007.7.1963

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The homotopy infinite symmetric product represents stable homotopy

Christian Schlichtkrull

Algebraic & Geometric Topology 7 (2007) 1963–1977

DOI: 10.2140/agt.2007.7.1963

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 $S P_{h} (X)$ 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

Volume 7, issue 4 (2007)