Idempotent functors that preserve cofiber sequences and split suspensions

Strom, Jeffrey

doi:10.2140/agt.2013.13.2335

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Idempotent functors that preserve cofiber sequences and split suspensions

Jeffrey Strom

Algebraic & Geometric Topology 13 (2013) 2335–2346

DOI: 10.2140/agt.2013.13.2335

Abstract

We show that an $f$ –localization functor $L_{f}$ commutes with cofiber sequences of $(N - 1)$ –connected finite complexes if and only if its restriction to the collection of $(N - 1)$ –connected finite complexes is $R$ –localization for some unital subring $R \subseteq ℚ$ . This leads to a homotopy theoretical characterization of the rationalization functor: the restriction of $L_{f}$ to simply connected spaces (not just the finite complexes) is rationalization if and only if $L_{f} (S^{2})$ is nontrivial and simply connected, $L_{f}$ preserves cofiber sequences of simply connected finite complexes and for each simply connected finite complex $K$ , there is a $k$ such that $Σ^{k} L_{f} (K)$ splits as a wedge of copies of $L_{f} (S^{n})$ for various values of $n$ .

Keywords

localization, rationalization, suspension

Mathematical Subject Classification 2010

Primary: 55P60, 55P62

Secondary: 55P35, 55P40

References

Publication

Received: 21 October 2012

Revised: 15 February 2013

Accepted: 18 February 2013

Published: 30 June 2013

Authors

Jeffrey Strom
Department of Mathematics Western Michigan University 1903 W. Michigan Ave. Kalamazoo, MI 49008 USA
http://homepages.wmich.edu/~jstrom/

Volume 13, issue 4 (2013)