Hurewicz Fibration

Definition

A map of topological spaces $p:E\to B$ is a Hurewicz fibration if it has the homotopy lifting property with respect to all spaces: for every space $X$, every homotopy $H:I\times X\to B$, and every lift $f:X\to E$ with $p\circ f=H(0,-)$, there exists a lift $\tilde{H}:I\times X\to E$ such that $p\circ \tilde{H}=H$ and $\tilde{H}(0,-)=f$.

Equivalently, $p$ has the covering homotopy property for maps $X\to B$ from arbitrary spaces $X$. In particular, paths and homotopies in the base lift to the total space compatibly with endpoints.

Properties

• Stability: closed under pullback, composition, and retracts.
• Long exact sequence: a Hurewicz fibration sequence $F\hookrightarrow E\stackrel{p}{\to }B$ induces the long exact sequence of homotopy groups relating $F$, $E$, and $B$.
• Path lifting: every path $\gamma :I\to B$ with a chosen lift of its initial point lifts to a path $\tilde{\gamma }:I\to E$ starting at the chosen point.
• Comparison: every Fibre Bundle is a Hurewicz fibration; every Hurewicz fibration is a Serre Fibration. The converses need not hold.

Examples

• Path-space fibration: evaluation at 1, ${\mathrm{ev}}_1:X^I\to X$, with fibre $\Omega X$.
• Trivial product: ${\mathrm{pr}}_1:B\times F\to B$.
• Locally trivial bundles: any fibre bundle (e.g. vector bundles, principal $G$-bundles).
• Covering spaces: $p:\tilde{B}\to B$ with discrete fibre.

Related Concepts

Topological Fibration Serre Fibration Fibre Bundle Kan Fibration Fibration

References

• https://ncatlab.org/nlab/show/Hurewicz+fibration
• hatcher2001-algebraic-topology