#### Volume 13, issue 4 (2013)

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Idempotent functors that preserve cofiber sequences and split suspensions

### Jeffrey Strom

Algebraic & Geometric Topology 13 (2013) 2335–2346
##### Abstract

We show that an $f$–localization functor ${L}_{f}$ commutes with cofiber sequences of $\left(N-1\right)$–connected finite complexes if and only if its restriction to the collection of $\left(N-1\right)$–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 ${L}_{f}\left({S}^{2}\right)$ 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 ${\Sigma }^{k}{L}_{f}\left(K\right)$ splits as a wedge of copies of ${L}_{f}\left({S}^{n}\right)$ for various values of $n$.

##### Keywords
localization, rationalization, suspension
##### Mathematical Subject Classification 2010
Primary: 55P60, 55P62
Secondary: 55P35, 55P40