Idempotent functors that preserve cofiber sequences and split suspensions

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 $R\subseteq \mathbb{Q}$. 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 $n$.