nLab Dold fibration

Serre fibration$\Leftarrow$ Hurewicz fibration $\Rightarrow$ Dold fibration$\Leftarrow$ shrinkable map

Idea

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

Definition

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

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

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 $\hat{g}(-,0):Y\to E$ 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 $H(y,t)=H(y,0)$ for $0\leq t\leq t_0$ for some $t_0\gt 0$. 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.

Relation to other fibrations

Proposition

Every Hurewicz fibration is a Dold fibration.

Proposition

Not all Serre fibrations are Dold fibrations.

David Roberts: Can we come up with a counterexample of a Serre fibration that isn’t a Dold fibration? I’ll ask on MathOverflow.

Proposition

Not all Dold fibrations are Serre fibrations.

Proof

Here is a very simple counter-example due to Dold. Consider the union of line segments

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

in $\mathbf{R}^2$, and the map projecting on to the first coordinate, $pr_1:E \to [-1,1]$. Then this map is a Dold fibration but not a Serre fibration.

Chris Schommer-Pries: I believe this is an example of a quasi-fibration which is not a Serre fibration, but it is not a Dold fibration either.

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.

Examples

Observation

Every shrinkable map is a Dold fibration.

This result follows from a theorem of Dold about locally homotopy trivial map?s being the same as Dold fibrations. It should be obvious that a shrinkable map is globally homotopy trivial, with trivial fibre.

Corollary

Let $\{U_i \to X\}$ be a numerable open cover. Let $C(\{U_i\})$ the Cech nerve (a simplicial object in topological spaces) and $|C(\{U_i\})| \in Top$ its geometric realization. Then the canonical map

$$|C(\{U_i\})| \to X$$

is shrinkable, hence a Dold fibration.

This observation is due to Segal. See shrinkable map.