Serre fibration

Builds on Homotopy lifting property Continuous function Topological space

Definition

Let $E$ and $B$ be topological spaces and let $p \colon E \to B$ be a continuous map. Let $I = [0, 1]$ . The map $p$ is a Serre fibration if it has the homotopy lifting property 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 is a Serre fibration, and every fiber bundle is a Serre fibration.

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Tag	`9KE5E84M3M`
URI	`m2://9KE5E84M3M`
Kind	definition
Writing	W0 stub
MSC	55R05
Defines	Serre fibration
Uses	Homotopy lifting property, Continuous function, Topological space
Changed	2 November 2026

Serre fibration

Definition

Remarks

Reading path

Further reading

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