## Algebraic & Geometric Topology

### Idempotent functors that preserve cofiber sequences and split suspensions

Jeffrey Strom

#### Abstract

We show that an $f$–localization functor $Lf$ 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 $R⊆ℚ$. This leads to a homotopy theoretical characterization of the rationalization functor: the restriction of $Lf$ to simply connected spaces (not just the finite complexes) is rationalization if and only if $Lf(S2)$ is nontrivial and simply connected, $Lf$ preserves cofiber sequences of simply connected finite complexes and for each simply connected finite complex $K$, there is a $k$ such that $ΣkLf(K)$ splits as a wedge of copies of $Lf(Sn)$ for various values of $n$.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 4 (2013), 2335-2346.

Dates
Revised: 15 February 2013
Accepted: 18 February 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715641

Digital Object Identifier
doi:10.2140/agt.2013.13.2335

Mathematical Reviews number (MathSciNet)
MR3073919

Zentralblatt MATH identifier
1276.55017

Subjects
Primary: 55P60: Localization and completion 55P62: Rational homotopy theory
Secondary: 55P35: Loop spaces 55P40: Suspensions

#### Citation

Strom, Jeffrey. Idempotent functors that preserve cofiber sequences and split suspensions. Algebr. Geom. Topol. 13 (2013), no. 4, 2335--2346. doi:10.2140/agt.2013.13.2335. https://projecteuclid.org/euclid.agt/1513715641

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