# nLab Serre fibration

Contents

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 (the product topological space with the topological interval), along (for definiteness) the left endpoint inclusion.

###### Definition

A Serre fibration is a $J_{Top}$-injective morphism, def. , 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 $h \colon D^n \times I \to X$ 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 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. ).

###### 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. 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). Theorem-page at a Serre fibration between CW-complexes is a Hurewicz fibration.

### 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. ).

###### 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. , 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 and prop. , 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 $\iota_\ast$.

(…)

long exact sequence of homotopy groups

(…)

### Local recognition

###### Lemma

(map is Serre fibration if it is locally so)

If $p\colon X \to B$ is a continuous function and $\mathcal{U} = \big\{ U_i \overset{}{\hookrightarrow} B \big\}_{i \in I}$ is an open cover of its codomain, such that the restriction of $p$ to each $U_i$ (i.e. its pullback to the cover) is a Serre fibration, then $p$ itself is a Serre fibration:

$\array{ X_{\vert \mathcal{U}} &\longrightarrow& X \\ {}^{{}_{\mathllap{ {Serre} \atop {fibration} }}} \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow {}^{{}_{\mathrlap{ p }}} \\ \underset{i \in I}{\sqcup} U_i &\underset{ }{\longrightarrow}& B } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \array{ X \\ {}^{{}_{\mathllap{ {Serre} \atop {fibration} }}} \big\downarrow {}^{{}_{\mathrlap{ p }}} \\ B }$

###### Remark

An analogous local recognition holds for Hurewicz fibrations but with numerable open covers. See there.

## Examples

### Empty bundles

###### Example

(empty bundles are Serre fibrations)
All empty bundles $\varnothing \longrightarrow B$ are Serre fibrations, because none of the commuting squares that one would have to lift in actually exist:

$\array{ D^{n} &\overset{ \not \exists }{\longrightarrow}& \varnothing \\ \big\downarrow && \big\downarrow \\ D^n \times I &\longrightarrow& B \mathrlap{\,,} }$

since the empty topological space is a strict initial object: There is no morphism to it from any inhabited space.

### Fiber bundles

###### Example

(fiber bundles are Serre fibrations)
Every topological fiber bundle (i.e. locally trivial, meaning that on some open cover it becomes a product projection) is a Serre fibration.

This follows immediately from Lemma .

Notice that the empty bundles of Example are examples of locally trivial fiber bundles (with typical fiber the empty topological space).

### Covering spaces

###### Example

(covering space projection is Serre fibration)

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

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

• Stephen Mitchell, Notes on Serre fibrations, 2001 (pdf, pdf)

• Tammo tom Dieck, Algebraic topology. European Mathematical Society, Zürich (2008) (doi:10.4171/048)

For more see at classical model structure on topological spaces.