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Dold fibration

Context

Topology

topology (point-set topology, point-free topology)

see also algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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)