nLab Serre fibration

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

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 $\iota_\ast$.

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long exact sequence of homotopy groups

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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:

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

See tom Dieck 08, p. 130, Thm. 6.3.3.

Remark

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

Examples

General

Example

The topological realization of a Kan fibration of simplicial sets is a Serre fibration.

(due to Quillen 1968, see also Goerss & Jardine 1999, Thm. 10.10)

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.).

Related concepts

• transport

• Serre spectral sequence

• classical model structure on topological spaces

References

• Daniel Quillen, The geometric realization of a Kan fibration is a Serre fibration, Proc. AMS 19 (1968) 1499-1500 [doi:1968-019-06/S0002-9939-1968-0238322-1, pdf]

• 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)

• Paul Goerss, J. F. Jardine: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999), Modern Birkhäuser Classics (2009) [doi:10.1007/978-3-0346-0189-4, webpage]

For more see at classical model structure on topological spaces.