nLab Serre fibration

Context

Topology

topology

algebraic topology

Examples

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

Contents

Definition

Definition

Write

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

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

Definition

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

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

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

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

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

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 $J_{Top}$-relative cell complexes (def. 1).

Proof

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

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 $X$ a finite CW-complex, then its inclusion $X \overset{(id, \delta_0)}{\longrightarrow} X \times I$ into its standard cylinder is a $J_{Top}$-relative cell complex (def. 1).

Proof

First erect a cylinder over all 0-cells

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

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

Proposition

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

$F_y \overset{\iota}{\hookrightarrow} X \overset{f}{\longrightarrow} Y$

for the fiber inclusion over that point. Then for every choice $x \colon \ast \to X$ of lift of the point $y$ through $f$, the induced sequence of homotopy groups

$\pi_{\bullet}(F_y, x) \overset{\iota_\ast}{\longrightarrow} \pi_\bullet(X, x) \overset{f_\ast}{\longrightarrow} \pi_\bullet(Y)$

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

$ker(f_\ast) \simeq im(\iota_\ast) \,.$
Proof

It is clear that the image of $\iota_\ast$ is in the kernel of $f_\ast$ (every sphere in $F_y\hookrightarrow X$ becomes constant on $y$, hence contractible, when sent forward to $Y$).

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

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

where $\kappa$ is the contracting homotopy witnessing that $f_\ast[\alpha] = 0$.

Now since $x$ is a lift of $y$, there exists a left homotopy

$\eta \;\colon\; \kappa \Rightarrow const_y$

as follows:

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

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

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

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

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

$\tilde \eta \;\colon\; S^{n-1} \times I \longrightarrow X \,.$

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

$[\alpha'] = [\alpha] \,.$

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

$\array{ \alpha' \colon & S^{n-1} &\overset{(id,0)}{\longrightarrow}& S^{n-1}\times I &\overset{\tilde \eta}{\longrightarrow}& X \\ & \downarrow^{\iota_n} && \downarrow^{\mathrlap{(\iota_n,id)}} && \downarrow^{\mathrlap{f}} \\ & D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ & \downarrow && && \downarrow \\ & \ast && \overset{y}{\longrightarrow} && Y } \,.$

The commutativity of the outer square says that $f_\ast \alpha'$ is constant, hence that $\alpha'$ is entirely contained in the fiber $F_y$. Said more abstractly, the universal property of fibers gives that $\alpha'$ factors through $F_y\overset{\iota}{\hookrightarrow} X$, hence that $[\alpha'] = [\alpha]$ is in the image of $\iota_\ast$.

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long exact sequence of homotopy groups

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Examples

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

Revised on April 13, 2016 05:24:02 by Urs Schreiber (131.220.202.202)