nLab
Serre fibration

Context

Topology

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

see also differential topology, 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

Definition

Definition

Write

J Top{D n(id,δ 0)D n×I} nMor(Top) J_{Top} \coloneqq \left\{ D^n \stackrel{(id,\delta_0)}{\hookrightarrow} D^n \times I \right\}_{n \in \mathbb{N}} \;\subset Mor(Top)

for the set of inclusions of the topological n-disks, into their cylinder objects (the product topological space with the topological interval), along (for definiteness) the left endpoint inclusion.

Definition

A Serre fibration is a J TopJ_{Top}-injective morphism, def. 1, hence a continuous function f:XYf \colon X \longrightarrow Y that has the right lifting property with respect to all inclusions of the form (id,0):D nD n×I(id,0) \colon D^n \hookrightarrow D^n \times I that include the standard topological n-disk into its standard cylinder object.

I.e. ff is a Serre fibration if for every commuting square of continuous functions of the form

D n X (id,0) f D n×I Y \array{ D^n &\longrightarrow& X \\ {}^{\mathllap{(id,0)}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^n \times I &\longrightarrow& Y }

then there exists a continuous function h:D nXh \colon D^n \to X such as to make a commuting diagram of the form

D n X (id,0) h f D n×I Y \array{ D^n &\longrightarrow& X \\ {}^{\mathllap{(id,0)}}\downarrow &{}^h\nearrow& \downarrow^{\mathrlap{f}} \\ D^n \times I &\longrightarrow& Y }
Remark

The class of Serre fibrations serves as the class of abstract fibrations in the classical model structure on topological spaces (whence “Serre-Quillen model structure”).

Properties

Closure properties

Proposition

A Serre fibration has the right lifting property against all retracts of J TopJ_{Top}-relative cell complexes (def. 1).

Proof

By general closure properties of projective and injective morphisms, see there this proposition for details.

Relation to Hurewicz fibrations

Remark

The condition in def. 2 is part of the condition on a Hurewicz fibration, hence every Hurewicz fibration is in particular a Serre fibration.

The converse is false:

Example

(Serre fibrations which are not Hurewicz fibrations)

An example of a generalized covering space which is a Serre fibration but not a Hurewicz fibration is given by Jeremy Brazas here.

But under some regularity condition it does becomes true:

Proposition

(Serre fibrations of CW-complexes are Hurewicz fibrations)

In the convenient category of compactly generated weakly Hausdorff topological spaces) a Serre fibration in which the total space and base space are both CW complexes is a Hurewicz fibration.

(No relationship between the covering map and the CW structures is required.)

This is due to (Steinberger-West 84) with the corrected proof due to (Cauty 92) (pointers via Peter May here).

Long exact sequences of homotopy groups

Since Serre fibrations are the abstract fibrations in the Serre-classical model structure on topological spaces, the following statement follows from general model category theory. But it may also be seen by direct inspection, as follows.

Lemma

For XX a finite CW-complex, then its inclusion X(id,δ 0)X×IX \overset{(id, \delta_0)}{\longrightarrow} X \times I into its standard cylinder is a J TopJ_{Top}-relative cell complex (def. 1).

Proof

First erect a cylinder over all 0-cells

xX 0D 0 X (po) xX 0D 1 Y 1. \array{ \underset{x \in X_0}{\coprod} D^0 &\longrightarrow& X \\ \downarrow &(po)& \downarrow \\ \underset{x\in X_0}{\coprod} D^1 &\longrightarrow& Y_1 } \,.

Assume then that the cylinder over all nn-cells of XX has been erected using attachment from J TopJ_{Top}. Then the union of any (n+1)(n+1)-cell σ\sigma of XX with the cylinder over its boundary is homeomorphic to D n+1D^{n+1} and is like the cylinder over the cell “with end and interior removed”. Hence via attaching along D n+1D n+1×ID^{n+1} \to D^{n+1}\times I the cylinder over σ\sigma is erected.

Proposition

Let f:XYf\colon X \longrightarrow Y be a Serre fibration, def. 2, let y:*Yy \colon \ast \to Y be any point and write

F yιXfY F_y \overset{\iota}{\hookrightarrow} X \overset{f}{\longrightarrow} Y

for the fiber inclusion over that point. Then for every choice x:*Xx \colon \ast \to X of lift of the point yy through ff, the induced sequence of homotopy groups

π (F y,x)ι *π (X,x)f *π (Y) \pi_{\bullet}(F_y, x) \overset{\iota_\ast}{\longrightarrow} \pi_\bullet(X, x) \overset{f_\ast}{\longrightarrow} \pi_\bullet(Y)

is exact, in that the kernel of f *f_\ast is canonically identified with the image of ι *\iota_\ast:

ker(f *)im(ι *). ker(f_\ast) \simeq im(\iota_\ast) \,.
Proof

It is clear that the image of ι *\iota_\ast is in the kernel of f *f_\ast (every sphere in F yXF_y\hookrightarrow X becomes constant on yy, hence contractible, when sent forward to YY).

For the converse, let [α]π (X,x)[\alpha]\in \pi_{\bullet}(X,x) be represented by some α:S n1X\alpha \colon S^{n-1} \to X. Assume that [α][\alpha] is in the kernel of f *f_\ast. This means equivalently that α\alpha fits into a commuting diagram of the form

S n1 α X f D n κ Y, \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y } \,,

where κ\kappa is the contracting homotopy witnessing that f *[α]=0f_\ast[\alpha] = 0.

Now since xx is a lift of yy, there exists a left homotopy

η:κconst y \eta \;\colon\; \kappa \Rightarrow const_y

as follows:

S n1 α X ι n f D n κ Y (id,1) id D n (id,0) D n×I η Y * y Y \array{ && S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ && D^n &\overset{\kappa}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{(id,1)}} && \downarrow^{\mathrlap{id}} \\ D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow \\ \ast && \overset{y}{\longrightarrow} && Y }

(for instance: regard D nD^n as embedded in n\mathbb{R}^n such that 0 n0 \in \mathbb{R}^n is identified with the basepoint on the boundary of D nD^n and set η(v,t)κ(tv)\eta(\vec v,t) \coloneqq \kappa(t \vec v)).

The pasting of the top two squares that have appeared this way is equivalent to the following commuting square

S n1 α X (id,1) f S n1×I (ι n,id) D n×I η Y. \array{ S^{n-1} &\longrightarrow& &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{(id,1)}}\downarrow && && \downarrow^{\mathrlap{f}} \\ S^{n-1} \times I &\overset{(\iota_n, id)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y } \,.

Because ff is a Serre fibration and by lemma 1 and prop. 1, this has a lift

η˜:S n1×IX. \tilde \eta \;\colon\; S^{n-1} \times I \longrightarrow X \,.

Notice that η˜\tilde \eta is a basepoint preserving left homotopy from α=η˜| 1\alpha = \tilde \eta|_1 to some αη˜| 0\alpha' \coloneqq \tilde \eta|_0. Being homotopic, they represent the same element of π n1(X,x)\pi_{n-1}(X,x):

[α]=[α]. [\alpha'] = [\alpha] \,.

But the new representative α\alpha' has the special property that its image in YY is not just trivializable, but trivialized: combining η˜\tilde \eta with the previous diagram shows that it sits in the following commuting diagram

α: S n1 (id,0) S n1×I η˜ X ι n (ι n,id) f D n (id,0) D n×I η Y * y Y. \array{ \alpha' \colon & S^{n-1} &\overset{(id,0)}{\longrightarrow}& S^{n-1}\times I &\overset{\tilde \eta}{\longrightarrow}& X \\ & \downarrow^{\iota_n} && \downarrow^{\mathrlap{(\iota_n,id)}} && \downarrow^{\mathrlap{f}} \\ & D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ & \downarrow && && \downarrow \\ & \ast && \overset{y}{\longrightarrow} && Y } \,.

The commutativity of the outer square says that f *αf_\ast \alpha' is constant, hence that α\alpha' is entirely contained in the fiber F yF_y. Said more abstractly, the universal property of fibers gives that α\alpha' factors through F yιXF_y\overset{\iota}{\hookrightarrow} X, hence that [α]=[α][\alpha'] = [\alpha] is in the image of ι *\iota_\ast.

(…)

long exact sequence of homotopy groups

(…)

Examples

Every locally trivial topological fiber bundle is in particular a Serre fibration.

In particular,

Example

(covering space projection is Serre fibration)

Every covering space projection is a Serre fibration, in fact a Hurewicz fibration (by this prop.).

References

  • M. Steinberger and J. West, Covering homotopy properties of maps between CW complexes or ANRs, Proc. Amer. Math. Soc. 92 (1984), 573-577.

  • R. Cauty, Sur les overts des CW-complexes et les fibrés de Serre, Colloquy Math. 63 (1992), 1–7

Revised on July 29, 2017 17:40:36 by Dexter Chua (193.60.93.97)