Note on a theorem of Dold on cofibrations. (English) Zbl 0645.55005

The main result of the paper is the following generalization of a theorem of Dold on cofibrations [A. Dold, Invent. Math. 6, 185-189 (1968; Zbl 0167.516)]. Consider the following commutative diagram in the category of topological spaces \[ \begin{matrix} E_ A &&& @>{i_ E}>> &&& E \\ & \nwarrow j_ a &&&& j\nearrow \\ p_ A\downarrow && D_ a & @>>{i_ D}> & D && \downarrow p \\ & \swarrow q_ a &&&& p\searrow \\ A&&& @>>i> &&& B \end{matrix} \] in which i and j are inclusions and the left triangle is thepullback of the right triangle over i such that \(D_ A=q^{- 1}(A)\) and \(E_ A=p^{-1}(A)\). Then the inclusion \(E_ A\cup D\subset E\) is a cofibration whenever (a) i is a closed cofibration, p is a fibration and j is a cofibration over B, or (b) i is a cofibration, p is a regular fibration and j is a closed cofibration over B. If in addition to (a) j is closed and a homotopy equivalence over B, then \(E_ A\cup D\) is a strong deformation retract of E. This result covers several theorems of A. Strøm [Math. Scand. 22, 130-142 (1968; Zbl 0181.265)] as special cases.
Reviewer: K.H.Kamps


55P05 Homotopy extension properties, cofibrations in algebraic topology
55R05 Fiber spaces in algebraic topology
55P10 Homotopy equivalences in algebraic topology
55R10 Fiber bundles in algebraic topology