## 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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