Difference between revisions of "Serre fibration"
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| − | + | 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|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 | with | ||
| − | + | $$f\mid_{K\times\{0\}} = p\circ F_0$$ | |
| − | |||
there is a mapping | 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 | ||
| + | {{Cite|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==== | ====References==== | ||
| − | + | {| | |
| − | + | |- | |
| − | + | |valign="top"|{{Ref|Se}}||valign="top"| J.P. Serre, "Homologie singulière des espaces fibrés. Applications" ''Ann. of Math.'', '''54''' (1951) pp. 425–505 {{MR|0045386}} {{MR|0039255}} {{MR|0039254}} {{ZBL|0045.26003}} | |
| − | + | |- | |
| − | + | |valign="top"|{{Ref|Sp}}||valign="top"| E.H. Spanier, "Algebraic topology", McGraw-Hill (1966) pp. Chapt. 2, §2; Chapt. 7, §2 {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} | |
| − | + | |- | |
| − | + | |} | |
Revision as of 22:24, 24 November 2013
2020 Mathematics Subject Classification: Primary: 15-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 |
How to Cite This Entry:
Serre fibration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serre_fibration&oldid=15485
Serre fibration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serre_fibration&oldid=15485
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article