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 $f:E\to B$ is said to have the weak covering homotopy property (WCHP) for the space $Y$ if for all squares

$$\begin{array}{ccc}Y\times \{0\}& \stackrel{g_0}{\to }& E\\ \downarrow & & \downarrow f\\ Y\times I& \underset{g}{\to }& B\end{array}$$\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 $\hat{g}:Y\times I\to E$ such that $f\circ \hat{g}=g$ and there is a vertical homotopy between $\hat{g}(-\mathrm{,0}):Y\to E$ 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 $H:Y\times I\to Z$ such that $H(y,t)=H(y\mathrm{,0})$ for $0\le t\le t_0$ 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 $\hat{g}:Y\times I\to E$ such that the diagram is strictly commutative. It is of course not required that $\hat{g}$ be delayed (one can require, but then one allows $t_0$ for $\hat{g}$ 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

$$E:=[-\mathrm{1,0}]\times \{2\}\cup \{0\}\times [\mathrm{1,2}]\cup [\mathrm{0,1}]\times \{1\}$$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, ${\mathrm{pr}}_1:E\to [-\mathrm{1,1}]$. 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.