Serre fibration

A Serre fibration is a map having the homotopy lifting property with respect to every disc, equivalently every CW complex.

For a based Serre fibration with fibre $F$ over the basepoint, lift a based cube in $B$ from its constant bottom face, keeping the side faces fixed. Its final face lies in $F$, giving surjectivity of the relative map. Homotopies in $B$ lift; two lifts of the same cube are relatively homotopic by lifting with both side liftings prescribed. This gives injectivity. For $n\ge 2$ this is a group isomorphism; at $n=1$ it is initially a bijection of pointed sets, unless the relative set is endowed with the group law transported through the fibration.

For a Serre fibration over $S^r$, cover the base by a contractible punctured sphere and a contractible ball. Excision relative to the preimage of the punctured sphere identifies the relative pair with the product of $F$ and $(D^r,S^{r-1})$ up to homotopy. The base pair has a single free homology group in degree $r$, so the Künneth theorem gives the displayed shift without a torsion term. The relative long exact sequence then gives $\cdots \to H_i(F)\to H_i(E)\to H_{i-r}(F)\to H_{i-1}(F)\to \cdots$. The connecting map retains the twisting of the original Serre fibration.

For a Serre fibration with this fiber over a simply connected CW base, the fiber identity class has only one possible nonzero outgoing differential in the cohomological Serre spectral sequence, landing in $H^{n+1}(B;G)$. If that group vanishes, the class extends to the CW total space. Its representing map, paired with projection to $B$, is an isomorphism on fiber and base homotopy groups and hence a weak homotopy equivalence to the product.

For a map $f:X\to B$, let $E_f$ consist of points of $X$ and paths starting at their images, with projection $p(x,\gamma )=\gamma (1)$. Appending segments of a base homotopy proves the homotopy lifting property, so $p$ is a Serre fibration. The constant-path inclusion $s:X\to E_f$ satisfies $ps=f$ and is a homotopy equivalence: forget the path for an inverse, and shorten paths to prove the homotopy. The homotopy need not preserve the endpoint in $B$, so the statement concerns an equivalence of total spaces over a commuting triangle, not a fibrewise inverse. The fibre over $b_0$ is the homotopy fibre of $f$.

A map $p:E\to B$ has the homotopy lifting property with respect to $X$ when every homotopy $X\times I\to B$ whose initial map lifts to $E$ has a lift extending that initial lift.

For a based fibration $F\to E\to B$, lifting representatives defines connecting maps and an exact sequence