Serre Fibration

Definition

A map of topological spaces $p:E\to B$ is a Serre fibration if it has the homotopy lifting property with respect to all CW-complexes. Equivalently, it has the right lifting property against the inclusions $D^n\hookrightarrow D^n\times I$ (as $x\mapsto (x,0)$) for all $n\ge 0$: for every commutative square

$$\begin{array}{rlr}{D}^n& ⟶E& \\ \downarrow & & \downarrow p\\ D^n\times I& ⟶B& \end{array}$$

there exists a lift $D^n\times I\to E$ making both triangles commute.

Every Hurewicz Fibration is a Serre fibration, but not conversely.

Properties

• Stability: closed under pullback, composition, and retracts.
• Long exact sequence: a fibration sequence $F\hookrightarrow E\stackrel{p}{\to }B$ induces the long exact sequence of homotopy groups relating $F$, $E$, and $B$ (with chosen basepoints).
• Path and homotopy lifting: paths and homotopies in $B$ lift along $p$ relative to endpoints.
• Model-categorical status: in the classical Quillen model structure on suitable categories of spaces (e.g. compactly generated weak Hausdorff spaces), fibrations are precisely the Serre fibrations and weak equivalences are the weak homotopy equivalences.

Examples

• Fibre bundles: every Fibre Bundle (e.g. vector bundles, principal $G$-bundles) is a Serre fibration.
• Covering spaces: $p:\tilde{B}\to B$ with discrete fibre.
• Path-space fibration: evaluation at 1, ${\mathrm{ev}}_1:X^I\to X$; in particular a Hurewicz (hence Serre) fibration.
• Trivial product: projection ${\mathrm{pr}}_1:B\times F\to B$.

Related Concepts

• Topological Fibration
• Hurewicz Fibration
• Fibre Bundle
• Kan Fibration
• Fibration

References