Fibration: Difference between revisions

Latest revision as of 00:45, 25 December 2010

This article defines a property of continuous maps between topological spaces

Definition

A continuous map $p:E\to B$ of topological spaces is termed a fibration or is said to have the homotopy lifting property if it is surjective and, given any map $F:X\times I\to B$ and a map ${\tilde {f}}:X\to E$ such that $p({\tilde {f}}(x))=f(x,0)$ , there exists a map ${\tilde {F}}:X\times I\to E$ satisfying:

$p\circ {\tilde {F}}=F$
$F(x,0)={\tilde {f}}(x)$

This is dual to the notion of a cofibration.

Relation with other properties

Weaker properties

Weak fibration (also called Serre fibration)

@@ Line 3: / Line 3: @@
 ==Definition==
-A [[continuous map]] <math>p:E \to B</math> of [[topological space]]s is termed a '''fibration''' or is said to have the '''homotopy lifting property''' if, given any map <math>F:X \times I \to B</math> and a map <math>\tilde{f}: X \to E</math> such that <math>p(\tilde{f}(x)) = f(x,0)</math>, there exists a map <math>\tilde{F}:X \times I \to E</math> satisfying:
+A [[continuous map]] <math>p:E \to B</math> of [[topological space]]s is termed a '''fibration''' or is said to have the '''homotopy lifting property''' if it is ''surjective'' and, given any map <math>F:X \times I \to B</math> and a map <math>\tilde{f}: X \to E</math> such that <math>p(\tilde{f}(x)) = f(x,0)</math>, there exists a map <math>\tilde{F}:X \times I \to E</math> satisfying:
 * <math>p \circ \tilde{F} = F</math>