nLab
Dold fibration

Idea

A Dold fibration is a map that allows lifting? of homotopies, with initial condition agreeing with a given map up to a vertical homotopy?.

Definition

The map f:EB is said to have the weak covering homotopy property (WCHP) for the space Y if for all squares

Y×{0} g 0 E f Y×I g B\begin{matrix} Y\times\{0\}& \stackrel{g_0}{\to} & E \\ \downarrow&&\, \downarrow f\\ Y\times I &\underset{g}{\to} & B \end{matrix}

there is a map ĝ:Y×IE such that fĝ=g and there is a vertical homotopy between ĝ(,0):YE and g 0. The synonymous expression weak homotopy lifting property (WHLP) is also used.

A continuous map is a Dold fibration if it has the WCHP for all spaces. Somewhat surprisingly, there is an equivalent condition in terms of delayed homotopies. A delayed homotopy is a homotopy H:Y×IZ such that H(y,t)=H(y,0) for 0tt 0 for some t 0>0. A continuous map is a Dold fibration iff in the diagram above in which g is a delayed homotopy, can be filled with a diagonal map ĝ:Y×IE such that the diagram is strictly commutative. It is of course not required that ĝ be delayed (one can require, but then one allows t 0 for ĝ to be possibly smaller than t 0 for g). This is sometimes called the delayed homotopy lifting property.

Examples and counter-examples

Any Hurewicz fibration is a Dold fibration

Serre fibrations are not Dold fibrations, and there is a very simple counter-example. Consider the union of line segments

E:=[1,0]×{2}{0}×[1,2][0,1]×{1}E:= [-1,0]\times\{2\} \cup \{0\}\times [1,2] \cup [0,1]\times\{1\}

in R 2, and the map projecting on to the first coordinate, pr 1:E[1,1]. Then this map is a Dold fibration but not a Serre fibration.

One could consider maps that have the WCHP for just cubes – these are a sort of hybrid Dold–Serre fibration (warning! nonstandard terminology. I just made it up, suggestions appreciated). For these maps there exists a long exact sequence in homotopy once basepoints are chosen. For classes of maps determined by (homotopy) lifting properties, this is about the minimum one needs to define such a long exact sequence. On the other hand, quasifibrations give rise to a long exact sequence in homotopy, but are defined by homotopy properties of the fibres.