Serre fibration

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





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, along (for definiteness) the left endpoint inclusion.


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 }

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


Closure properties


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


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.


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).


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.


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) \,.

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



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

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