---
tag: R7G4ZC0963
kind: definition
name: Localization of a space
msc: [55P60]
defines: D0ZHVD797D
uses: [P976WYEFMB, M0Q5CFJ7ZH, 5W95QFHXPR, 01826KMG2H, KET2GSPZFG]
sources:
  - url: https://en.wikipedia.org/wiki/Localization_of_a_topological_space
---

## Definition

Let $P$ be a set of [[prime numbers|01826KMG2H]].
Let $\Z_{(P)}$ be the [[localization|KET2GSPZFG]] of $\Z$ at $P$: the rational numbers whose denominators are divisible by no prime in $P$.
Let $X$ be a simply connected [[CW complex|5W95QFHXPR]].
The space $X$ is *$P$-local* if each [[homotopy group|P976WYEFMB]] $\pi_n(X)$, $n \ge 2$, is a $\Z_{(P)}$-module, that is, multiplication by every prime $q \notin P$ is bijective on it.
A *$P$-localization* of $X$ is a map $f \colon X \to X_P$ to a simply connected $P$-local space such that $f$ induces isomorphisms
$$\pi_n(X) \otimes \Z_{(P)} \to \pi_n(X_P) \quad \text{for all } n \ge 2.$$
For $P = \emptyset$ one has $\Z_{(P)} = \Q$, and $f$ is called a *rationalization*, written $X \to X_\Q$.

## Remarks

Every simply connected CW complex has a $P$-localization, unique up to homotopy equivalence under $X$.
The condition on homotopy groups is equivalent to: $f$ induces isomorphisms $\tilde H_n(X; \Z) \otimes \Z_{(P)} \to \tilde H_n(X_P; \Z)$ in reduced [[singular homology|M0Q5CFJ7ZH]] for all $n$.
The localization is universal: every map from $X$ to a $P$-local space factors through $f$, uniquely up to homotopy.
The theory extends to nilpotent spaces.

Example: for $n$ odd, the map $S^n \to K(\Q, n)$ that represents a generator of $H^n(S^n; \Q)$ is a rationalization.
Indeed, $\pi_k(S^n) \otimes \Q$ is $\Q$ for $k = n$ and $0$ otherwise, by [[Serre's finiteness theorem|Y40ZRJJNH0]], which matches the homotopy of $K(\Q, n)$.

Non-example: the identity of $S^2$ is not a rationalization, because $S^2$ is not $P$-local for $P = \emptyset$: $\pi_2(S^2) \cong \Z$ is not a $\Q$-vector space.