nLab
Dold fibration

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

Contents

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:EBf:E \to B of topological spaces is said to have the weak covering homotopy property (WCHP) for the space YY 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 g^:Y×IE\hat{g}:Y\times I \to E such that fg^=gf\circ \hat{g} = g and there is a vertical homotopy between g^(,0):YE\hat{g}(-,0):Y\to E and g 0g_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×IZH:Y\times I\to Z such that H(y,t)=H(y,0)H(y,t)=H(y,0) for 0tt 00\leq t\leq t_0 for some t 0>0t_0\gt 0. A continuous map is a Dold fibration iff in the diagram above in which gg is a delayed homotopy, can be filled with a diagonal map g^:Y×IE\hat{g}:Y\times I \to E such that the diagram is strictly commutative. It is of course not required that g^\hat{g} be delayed (one can require, but then one allows t 0t_0 for g^\hat{g} to be possibly smaller than t 0t_0 for gg). 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]×{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\mathbf{R}^2, and the map projecting on to the first coordinate, pr 1:E[1,1]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 iX}\{U_i \to X\} be a numerable open cover. Let C({U i})C(\{U_i\}) the Cech nerve (a simplicial object in topological spaces) and C({U i})Top|C(\{U_i\})| \in Top its geometric realization. Then the canonical map

C({U i})X |C(\{U_i\})| \to X

is shrinkable, hence a Dold fibration.

This observation is due to Segal. See shrinkable map.

Revised on June 25, 2013 23:55:04 by Schommer Pries? (88.153.13.194)