Regular fibrations revisited. (English) Zbl 0604.55007

[Note: Some diagrams below cannot be displayed correctly in the web version. Please use the PDF version for correct display.]
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


55R05 Fiber spaces in algebraic topology
55P05 Homotopy extension properties, cofibrations in algebraic topology
Full Text: DOI


[1] R. Brown andP. R. Heath, Coglueing homotopy equivalences. Math. Z.113, 313-325 (1970). · Zbl 0185.51101
[2] T.Tom Dieck, K. H.Kamps und D.Puppe, Homotopietheorie. LNM157, Berlin 1970.
[3] J.Dugundji, Topology. Boston 1966.
[4] A. Strøm, Note on cofibrations. Math. Scand.19, 11-14 (1966). · Zbl 0145.43604
[5] A. Strøm, Note on cofibrations II. Math. Scand.22, 130-142 (1968). · Zbl 0181.26504
[6] A. Strøm, The homotopy category is a homotopy category. Arch. Math.23, 435-441 (1972). · Zbl 0261.18015
[7] P. Tulley, On regularity in Hurewicz fiber spaces. Trans. Amer. Math. Soc.116, 126-134 (1965). · Zbl 0142.21803
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.