---
tag: 9KE5E84M3M
kind: definition
name: Serre fibration
msc: [55R05]
defines: PB9JXQ2KC3
uses: [G4QDM01P6J, 0X3WGX95FE, 0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Fibration
---

## Definition

Let $E$ and $B$ be [[topological spaces|0CQHFPGJ45]] and let $p \colon E \to B$ be a [[continuous map|0X3WGX95FE]].
Let $I = [0, 1]$.
The map $p$ is a *Serre fibration* if it has the [[homotopy lifting property|G4QDM01P6J]] with respect to the cube $I^n$ for every integer $n \ge 0$.
That is, let $n \ge 0$, let $\tilde f \colon I^n \to E$ be continuous, and let $H \colon I^n \times I \to B$ be continuous with $H(x, 0) = p(\tilde f(x))$ for all $x$.
Then there is a continuous $\tilde H \colon I^n \times I \to E$ with $p \circ \tilde H = H$ and $\tilde H(x, 0) = \tilde f(x)$ for all $x \in I^n$.

## Remarks

Example: for spaces $B$ and $F$, the projection $p \colon B \times F \to B$ is a Serre fibration.
Given $\tilde f = (f_1, f_2)$ and $H$, the map $\tilde H(x, t) = (H(x, t), f_2(x))$ is a lift with the right start.

Non-example: the inclusion $p \colon \set{0} \to [0, 1]$ is not a Serre fibration.
Take $n = 0$, $\tilde f$ the map to $0$ and $H(t) = t$; any lift would satisfy $p(\tilde H(1)) = 1$, but the only point of $\set{0}$ maps to $0$.

Equivalently, $p$ has the homotopy lifting property with respect to all CW complexes.
Every [[Hurewicz fibration|Y773B7EQA7]] is a Serre fibration, and every [[fiber bundle|CX4M8K61AB]] is a Serre fibration.