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Regular fibrations revisited. (English) Zbl 0604.55007

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The results of this paper are variations and extensions of theorems of A. Strøm. A Hurewicz fibration \(p: E\to B\) is called regular if there is a lifting map \(\lambda: \Omega_ p\to E^ I\), where \(\Omega_ p:=\{(e,\omega)|\) \(e\in E\), \(\omega \in B^ I\), \(p(e)=\omega (0)\}\), such that \(\lambda(e,\omega)\) is a constant path whenever \(\omega\) is constant. It is shown: For a regular fibration \(p: E\to B\) the following properties of an inclusion \(i: A\subset B\) are transferred to \(j: p^{- 1}(A)\subset E:\) (a) cofibration; (b) weak cofibration; (c) strong deformation retraction. An example shows that the condition of regularity in case (b) cannot be omitted.
A Hurewicz fibration is called strongly regular if the associated fibration \(\tilde p: E^ I\to \Omega_ p\), \(\tilde p(e,\omega):= (\omega(0),p\circ \omega)\), is regular. The map p is said to have the lifting homotopy extension property (LHEP) with respect to \(i: A\subset X\) if any commutative square \[ \begin{tikzcd} X\times 0\cup A\times I \ar[r,"\phi"]\ar[d,"\tilde{\subset}" '] & E \ar[d,"p"]\\ X\times I \ar[r,"\Phi" '] & B \end{tikzcd} \] admits a diagonal \(X\times I\to E\). It is proved: A strongly regular fibration has LHEP with respect to all weak cofibrations \(i: A\subset X\). The converse of this is true if E and B are Hausdorff spaces.
Reviewer: W.End

MSC:

55R05 Fiber spaces in algebraic topology
55P05 Homotopy extension properties, cofibrations in algebraic topology
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References:

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