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For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.

Let $\pi : E \to B$ be a continuous map and $X$ a topological space.

We say that $\pi : E \to B$ has the homotopy lifting property with respect to $X$ if for any homotopy $f : X \times [0, 1] \to B$ and any map $F_0 : X \to E$ such that $\pi(F_0(x)) = f(x, 0)$, there is a homotopy $F : X\times [0, 1] \to E$ such that $\pi(F(x, t)) = f(x, t)$ and $F(x, 0) = F_0(x)$.

We say that $\pi : E \to B$ is a fibration if it has the homotopy lifting property with respect to $X$ for all topological spaces $X$. If $\pi : E \to B$ has the homotopy lifting property for all CW complexes, then it is called a Serre fibration.

If $\pi : E \to B$ is a Serre fibration and $B$ is path-connected, then the homotopy type of $\pi^{-1}(b)$ is independent of $b$; we denote this by $F$ and refer to it as the fiber of the fibration which can now be written as $F \to E \xrightarrow{\pi} B$.

Given a Serre fibration, there is an associated long exact sequence is homotopy

$$\dots \to \pi_{n+1}(B) \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \dots$$

The dual notion of a fibration is the notion of a .