Idempotent functors that preserve cofiber sequences and split suspensions

TL;DR

This paper characterizes when localization functors commute with cofiber sequences and split suspensions, providing a homotopy-theoretical criterion for rationalization in terms of their behavior on spheres and finite complexes.

Contribution

It offers a new homotopy-theoretical characterization of rationalization functors based on their preservation of cofiber sequences and splitting properties.

Findings

Localization functors commuting with cofiber sequences are characterized by R-localization.

Rationalization is characterized by specific conditions on the localization of spheres and finite complexes.

Splitting of suspensions of finite complexes into wedges of spheres is key to the characterization.

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 $R\sseq\mathbb{Q}$. 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 $L_f(S^2)$ is nontrivial and simply-connected, $L_f$ preserves cofiber sequences of simply-connected finite complexes, and for each simply-connected finite complex $K$, $\s^k L_f(K)$ splits as a wedge of copies of $L_f(S^n)$ for large enough $k$ and various values of $n$.