Serre fibration

2020 Mathematics Subject Classification: Primary: 55-XX [MSN][ZBL]

A triple $(X,p,Y)$, where $X$ and $Y$ are topological spaces and $p:X\to Y$ is a continuous mapping, with the following property (called the property of the existence of a covering homotopy for polyhedra). For any finite polyhedron $K$ and for any mappings

$$f:K\times[0,1]\to Y, \qquad F_0:K=K\times\{0\}\to X$$ with

$$f\mid_{K\times\{0\}} = p\circ F_0$$ there is a mapping

$$F : K\times[0,1]\to X$$ such that $F\mid_{K\times\{0\}} = F_0$, $p\circ F=f$. It was introduced by J.-P. Serre in 1951 (see [Se]).

Comments

A Serre fibration is also called a weak fibration. If the defining homotopy lifting property holds for every space (not just polyhedra), $p:X\to Y$ is called a fibration or Hurewicz fibre space.

References

[Se]	J.P. Serre, "Homologie singulière des espaces fibrés. Applications" Ann. of Math., 54 (1951) pp. 425–505 MR0045386 MR0039255 MR0039254 Zbl 0045.26003
[Sp]	E.H. Spanier, "Algebraic topology", McGraw-Hill (1966) pp. Chapt. 2, §2; Chapt. 7, §2 MR0210112 MR1325242 Zbl 0145.43303