Fibrations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 9 September 2012

Introduction

Cofibrations and Fibrations are analogues of exact sequences in Top and ${\text{Top}}_{✶}$.

Let $\iota :A⟶X$ be a morphism and let $Y\in \text{Top}$.

The morphism $\iota :A⟶X$ has the homotopy extension property with respect to $Y$ if $\iota$ satisfies:

$\begin{array}{c}\text{if}f:X⟶Y\text{and}h:A⟶\text{Hom}([0,1],Y)\text{are morphisms such that}\\ \text{if}a\in A\text{then}\left(h\left(a\right)\right)\left(0\right)=f\left(\iota \left(a\right)\right)\\ \text{then there exists}\stackrel{\sim }{h}:X⟶\text{Hom}([0,1],Y)\text{such that}\\ \text{if}x\in X\text{then}\left(\stackrel{\sim }{h}\left(x\right)\right)\left(0\right)=f\left(x\right)\text{.}\end{array}$

A cofibration is a morphism $\iota :A⟶X$ such that if mat y∈Top then $\iota :A⟶X$ has the homotopy extension property with respect to $Y$.

The cofibre of a cofibration $\iota :A⟶X$ is ${\displaystyle \frac{X}{\iota \left(X\right)}}$ and

$\begin{array}{ccccc}A& \stackrel{\iota }{⟶}& X& \stackrel{p}{⟶}& {\displaystyle \frac{X}{\iota \left(A\right)}}\end{array}$

is the cofibration sequence of $\iota :A⟶X$.

Let $f:X⟶Y$ be a morphism.

The mapping cone of $f$ is the pushout $\text{Con}\left(f\right)$,

and ${\iota }_0:X⟶CX$ is given by ${\iota }_0\left(x\right)=(x,0)$.

The mapping cylinder of $f$, or homotopy cofibre of $f$, is the pushout $\text{Cyl}\left(f\right)$.

$\begin{array}{ccc}\begin{array}{ccc}X& \stackrel{f}{⟶}& Y\\ {\iota }_0\downarrow & & \downarrow \\ X\times [0,1]& ⟶& \text{Cyl}\left(f\right)\end{array}& ,\text{where}& \begin{array}{cccc}{\iota }_0:& X& ⟶& X\times [0,1]\\ & x& ⟼& (x,0)\end{array}\end{array}$

Let $f:X⟶Y$ be a morphism.

1. Then $f=\tau \circ j$

   $\begin{array}{cc}\begin{array}{cccccc}f:& X& \stackrel{j}{⟶}& \text{Cyl}\left(f\right)& \stackrel{\tau }{⟶}& Y\\ & x& ⟼& (x,1)\\ & & & y& ⟼& y\\ & & & (x,t)& ⟼& f\left(x\right)\end{array}& \text{and}\tau \text{is a homotopy equivalence.}\end{array}$

2. The morphism $j:X⟶\text{Cyl}\left(f\right)$ is a cofibration with cofibration sequence

   $\begin{array}{ccccc}X& \stackrel{j}{⟶}& \text{Cyl}\left(f\right)& ⟶& \text{Con}\left(f\right)\end{array}$

Let $p:E⟶B$ be a morphism and let $Y\in \text{Top}$.

The morphism $p:E⟶B$ has the homotopy lifting property with respect to $Y$ if $p$ satisfies:

$\begin{array}{c}\text{if}f:Y⟶E\text{and}h:Y\times [0,1]⟶B\text{are morphisms such that}\\ \text{if}y\in Y\text{then}h(y,0)=p\left(f\left(y\right)\right)\\ \text{then there exists}\stackrel{\sim }{h}:Y\times [0,1]⟶E\text{such that}\\ \text{if}(y,t)\in Y\times [0,1]\text{then}h(y,t)=p\left(\stackrel{\sim }{h}(y,t)\right)\end{array}$

A Hurewicz fibration is a morphism $p:E⟶B$ such that if $Y\in \text{Top}$ then $p:E⟶B$ satisfies the homotopy lifting property with respect to $Y$.

A fibration, or Serre fibration is a morphism $p:E⟶B$ such that,

if $Y\in CW$ then $p:E⟶B$ satisfies the homotopy lifting property with respect to $Y\text{.}$

The fibre of a fibration $p:E⟶B$ is $p^{-1}\left(b_0\right)$ and

$\begin{array}{ccccc}{p}^{-1}\left(b_0\right)& \stackrel{\iota }{⟶}& E& \stackrel{p}{⟶}& B\end{array}$

is the fibration sequence of $p:E⟶B$.

Let $f:X⟶Y$ be a morphism.

The homotopy fibre of $f$ is the pullback $F_f$,

$\begin{array}{ccc}\begin{array}{ccc}{F}_f& ⟶& X\\ \downarrow & & \downarrow f\\ {\text{Map}}_{✶}([0,1],y)& \underset{p_1}{⟶}& Y\end{array}& \text{where}& \begin{array}{cccc}{p}_1:& {\text{Map}}_{✶}([0,1],y)& ⟶& Y\\ & \beta & ⟼& \beta \left(1\right)\end{array}\end{array}$

The mapping path space of $f$ is the pullback $N_f$

$\begin{array}{ccc}\begin{array}{ccc}{N}_f& ⟶& \text{Map}([0,1],y)\\ \downarrow & & \downarrow p_1\\ X& \underset{f}{⟶}& Y\end{array}& \text{where}& \begin{array}{cccc}{p}_1:& \text{Map}([0,1],y)& ⟶& Y\\ & \beta & ⟼& \beta \left(1\right)\end{array}\end{array}$

Let $f:X⟶Y$ be a morphism.

1. Then $f=\rho \circ \nu$

   $\begin{array}{cc}\begin{array}{cccccc}f:& X& \stackrel{\nu }{⟶}& N_f& \stackrel{\rho }{⟶}& Y\\ & x& ⟼& (x,C_{f\left(x\right)})\\ & & & (x,\beta )& ⟼& \beta \left(0\right)\end{array}& \text{and}\nu \text{is a homotopy equivalence,}\end{array}$

   where, something, $C_y:[0,1]⟶Y$ is given by $C_y\left(t\right)=y$.

2. The morphism $\rho :N_f⟶Y$ is a fibration with fibre $p^{-1}\left(✶\right)=F_f$ and fibration sequence

   $\begin{array}{ccccc}{F}_f& ⟶& N_f& \stackrel{\rho }{⟶}& Y\end{array}$

HW: For a fibration we almost get an action of ${\pi }_1(B,b_0)$ on the fibre.

Covering spaces and universal covers

A covering space, or cover of $B$, is a fiber bundle with discrete fibers.

Let $B$ be a path connected space.

A universal cover of $B$ is a covering space $E\stackrel{p}{⟶}B$ such that

1. $E$ is path connected,
2. ${\pi }_1(E,e_0)=0$.

(Universal property) A universal cover $E\stackrel{p}{⟶}B$ satisfies the universal property
if $E^{\prime }\stackrel{p^{\prime }}{⟶}B$ is a cover of $B$ then there exists a unique $E\stackrel{h}{⟶}E$ such that $p=h\circ p^{\prime }$

$\begin{array}{ccc}E& \stackrel{h}{⟶}& E^{\prime }\\ p\downarrow & & \downarrow p^{\prime }\\ B& \underset{{\text{id}}_B}{⟶}& B\end{array}$

[Benson II, Theorem 1.16.13]. The map

$\begin{array}{ccc}\left\{\genfrac{}{}{0ex}{}{\text{conjugacy classes of}}{\text{subgroups}H\subseteq {\pi }_1(B,b_0)}\right\}& ⟶& \left\{\genfrac{}{}{0ex}{}{\text{isomorphism classes of}}{\text{covers}{E}^{\prime }\stackrel{p^{\prime }}{⟶}B}\right\}\\ H& ⟼& (E,e_0)/H\end{array}$

Nerves and classifying spaces

Let $𝒞$ be a category.

The nerve of $𝒞$ is the simplicial set $𝒩𝒞$ given by

$𝒩{𝒞}_n=\left\{\text{sequences}{M}_1\stackrel{f_1}{⟶}{M}_2\stackrel{f_2}{⟶}\dots \stackrel{f_n}{⟶}{M}_{n+1}\text{of morphisms in}𝒞\right\}$

The classifying space $\mathrm{Bscr;}𝒞$ of $𝒞$ is the topological realization of $𝒩𝒞$,

$\mathrm{Bscr;}𝒞=\mid 𝒩𝒞\mid .$

Let $G$ be a discrete group.

The Cayley category $𝒞\left(G\right)$ of $G$ has

$\text{objects:}g\in G\text{morphism:}\left\{g\stackrel{h}{⟶}hg\right\}=\text{Hom}(g,hg)$

The category of $G$ has

$\text{one object:}\cdot ,\text{and morphisms}\text{Hom}(\cdot ,\cdot )=G$

HW: Show that

Notes and References

For simplicial sets and the definition of the nerve and classifying space of a category, see [Benson II §1.8]. For the connection between nerves of categories and classifying spaces of groups see [Benson II, §2.4].

Examples of universal central extensions

1. For $n\ne 6,7,$

   $0⟶\mathbb{Z}/2\mathbb{Z}⟶{\stackrel{\sim }{A}}_n⟶A_n⟶1$

   is a universal central extension of $A_n$ (obtained by restricting the central extension

   $0⟶\mathbb{Z}/2\mathbb{Z}⟶{\text{Spin}}_{n-1}\left(ℝ\right)⟶S{O}_{n-1}⟶1.)$

2. For $q\ne 2,3,$

   $0⟶{\mu }_n\left({𝔽}_q\right)⟶S{L}_n\left({𝔽}_q\right)⟶PS{L}_n\left({𝔽}_q\right)⟶1$

   is the universal central extension of $PS{L}_n\left({𝔽}_q\right)$.

3. Let $n\ge 3$. Let $R$ be a ring and let $E_n\left(R\right)$ be the subgroup of $G{L}_n\left(R\right)$ generated by the elementary matrices $x_{ij}\left(v\right)$.

   The Steinberg group $S{t}_n\left(R\right)$ is given by generators $x_{ij}\left(r\right)$ for

   $r\in R,1\le i,j\le n,$

   with relations

   $\begin{array}{c}{x}_{ij}\left(r_1\right)x_{ih}\left(r_2\right)=x_{ij}(r_1+r_2)\\ [x_{ij}\left(r_1\right),x_{jk}\left(r_2\right)]=x_{ik}\left(r_1{r}_2\right),\text{for}i\ne k\\ [x_{ij}\left(r_1\right),x_{kl}\left(r_2\right)]=1,\text{for}j\ne k\text{and}i\ne l\text{.}\end{array}$

   Then, for $n\ge 5$,

   $1⟶K_2(u,R)⟶S{t}_n\left(R\right)⟶E_n\left(R\right)⟶1$

   is a universal central extension of $E_n\left(R\right)$.

   The Milnor $K$–group is

   $K_2\left(R\right)=\underset{⟶}{\text{lim}}{K}_2(n,R)\text{.}$

Notes and References

These examples of universal central extensions are taken from [Weibel, §6.9].

Examples of universal covers and principal bundles

1. $ℝ⟶S^1$ is a universal cover with ${\pi }_1(S^1,s_0)=\mathbb{Z}$
2. $S^n⟶ℝP^n$ is a universal cover with ${\pi }_1(S^1\times S^1,x_0)=\mathbb{Z}/2$
3. $R^2⟶S^1\times S^1$ is a universal cover with ${\pi }_1(S^1\times S^1,x_0)=\mathbb{Z}\times \mathbb{Z}$
4. $S^n⟶ℝP^n$ is a principal $\mathbb{Z}/2\mathbb{Z}$–bundle. $(\mathbb{Z}/2\mathbb{Z}=O\left(1\right)=O_1\left(ℝ\right))$
5. $S^{\infty }⟶ℝP^{\infty }$ is a universal principal $\mathbb{Z}/2\mathbb{Z}$–bundle.
6. $ℝ⟶S^1$ is a universal principal $\mathbb{Z}$–bundle.
7. $0⟶\mathbb{Z}\stackrel{2\pi i}{⟶}ℂ\stackrel{\text{exp}}{⟶}{ℂ}^{\times }⟶1$ is a universal principal $\mathbb{Z}$–bundle.
8. $S^{2n+1}⟶ℂP^n$ is a principal $S^1$–bundle. $(S^1=U\left(1\right))$
9. $S^{\infty }⟶ℂP^{\infty }$ is a universal principal $S^1$–bundle.
10. Let $V_k\left({ℝ}^n\right)=\left\{\text{orhonormal}k\text{–frames in}{ℝ}^n\right\}$ be the Stiefel manifold, and

    $G_k\left({ℝ}^n\right)=\left\{k\text{dimensional subspaces in}{ℝ}^n\right\}\text{the}$

    Grassmannian of $k$–planes in ${ℝ}^n$.

    $V_k\left({ℝ}^n\right)⟶G{r}_k\left({ℝ}^n\right)\text{is a principal}O\left(k\right)\text{–bundle.}$

11. $V_k\left({ℝ}^{\infty }\right)⟶G_k\left({ℝ}^{\infty }\right)\text{is a universal principal}O\left(k\right)\text{–bundle.}$
12. Spheres:

    $\begin{array}{c}{S}^{n-1}\text{is a sphere in}{ℝ}^n,\\ S^{2n-1}\text{is a sphere in}{ℂ}^n,\\ S^{4n-1}\text{is a sphere in}{ℍ}^n,\end{array}$

    and

    $\begin{array}{c}SO(n-1)⟶SO\left(n\right)⟶S^{n-1}\text{is a principal}SO(n-1)\text{bundle,}\\ \begin{array}{c}U(n-1)⟶U\left(n\right)⟶S^{2n-1}\text{is a principal}U(n-1)\text{bundle,}\\ \begin{array}{c}SU(n-1)⟶SU\left(n\right)⟶S^{2n-1}\text{is a principal}SU(n-1)\text{bundle,}\\ \begin{array}{c}Sp(n-1)⟶Sp\left(n\right)⟶S^{4n-1}\text{is a principal}Sp(n-1)\text{bundle,}\end{array}\end{array}\end{array}\end{array}$

    where in each case the map is given by

    $A⟼A_{e_n}\text{where}{e}_n=(0,0,\dots ,0,1)\text{in}{ℝ}^n,{ℂ}^n\text{or}{ℍ}^n\text{.}$

13. Hopf fibrations Let $𝕆$ be the division ring of octonions.

    $ℝP^1=S^1,ℂP^1=S^2,ℍP^1=S^4,𝕆P^1=S^8$

    and the $n=2$ case of the principal bundles in $\left(ℍ\right)$ are

    $\begin{array}{ccc}(S^1⟶S^1=ℝP^1)& =& (U_1\left(ℝ\right)⟶SO\left(2\right)⟶ℝP^1)\\ (S^3⟶S^2=ℂP^1)& =& (U_1\left(ℂ\right)⟶SU\left(2\right)⟶ℂP^1)\\ (S^7⟶S^4=ℍP^1)& =& (U_1\left(ℍ\right)⟶S{U}_2\left(ℍ\right)⟶ℍP^1)\\ (S^{15}⟶S^8=𝕆P^1)& =& (U_1\left(𝕆\right)⟶S{U}_2\left(𝕆\right)⟶𝕆P^1)\end{array}$

    where the map is $v⟼\text{span}\left(v\right)$ in each case.

14. Projective Spaces

    O(n-1) ⟶ SO(n) ⟶ ℝPn-1 U(n-1) ⟶ SU(n) ⟶ ℂPn-1

    are principal bundles.

15. Grassmannians: The Grassmannian of $k$–dimensional subspaces of ${ℝ}^n$ is

    $G_k\left({ℝ}^n\right)=\left\{k\text{dimensional subspaces of}{ℝ}^n\right\}\text{.}$

    The sequences

    $\begin{array}{cccc}O\left(k\right)\times O(n-k)& ⟶& O\left(n\right)& ⟶G_k\left({ℝ}^n\right)\\ U\left(k\right)\times U(n-k)& ⟶& U\left(n\right)& ⟶G_k\left({ℂ}^n\right)\\ S_p\left(k\right)\times S_p(n-k)& ⟶& S_p\left(n\right)& ⟶G_k\left({ℍ}^n\right)\end{array}$

    are principal bundles.

16. The Steifel manifold is

    $V_k\left({ℝ}^n\right)=\left\{\text{orthonormal}k\text{–frames in}{ℝ}^n\right\}$.

    The sequence

    $O(n-k)⟶O\left(n\right)⟶V_k\left({ℝ}^n\right)$

    is a principal bundle.

17. $\begin{array}{ccc}{S}^1& ⟶& S^1\\ z& ⟼& z^n\end{array}$ is a principal $\mathbb{Z}/n\mathbb{Z}$ bundle.
18. Let $S$ be a Riemann surface of genus $g$.

    Assume $g\ne 0$.

    Then ${\pi }_1\left(S\right)$ is presented by generators

    $a_1,\dots ,a_g,b_1,\dots ,b_g$

    with relations

    $[a_1,b_1][a_2,b_2]\dots [a_g,b_g]=1\text{.}$

    Then

    $\mathscr{H}⟶S$ is a universal principal bundle for ${\pi }_1\left(S\right)$,

    where $\mathscr{H}$ is the hyperbolic plane.

19. Let $G$ be a connected Lie group, $K$ a maximal compact subgroup.

    If $\Gamma$ is a discrete torsion free subgroup of $G$ then

    ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.⟶\Gamma \setminus \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.}$ is a universal principal $\Gamma$–bundle.

20. let $G=S{L}_n\left(ℝ\right)$ and $K=S{O}_n\left(ℝ\right)$.

    For $N\ge 3$,

    $\Gamma \left(N\right)=\{\gamma \in S{L}_n\left(\mathbb{Z}\right)\mid \gamma =1\text{mod}N\}$

    is a discrete torsion free subgroup of $S{L}_n\left(ℝ\right)$. So

    $\raisebox{1ex}{$S{L}_n\left(ℝ\right)$}\!\left/ \!\raisebox{-1ex}{$S{O}_n\left(ℝ\right)$}\right.⟶\Gamma \left(N\right)\setminus \raisebox{1ex}{$S{L}_n\left(ℝ\right)$}\!\left/ \!\raisebox{-1ex}{$S{O}_n\left(ℝ\right)$}\right.$ is a principal $\Gamma \left(N\right)$–bundle.

Notes and References

These examples of covering spaces and principal bundles are taken from [Benson II, p21, 32, 36 and 41], [Weibel, p205], [McCleary p208] and [Bröcker-tomDieck p36-38].

Notes and References

Basic Theory of cobfibrations and fibrations is found in [May, Chapt. 6-8], [Benson, $$], [McClearly, p96 and p112] and [Weibel, p127].

The theorem that a fibre bundle is a Serre fibration is found in [Benson II, Theorem 1.6.11].

[May] J.P. May, A concise course in Algebraic Topology, ????.

Example

$\begin{array}{ccccc}\text{Let}& X& ⟶& X\times X& \text{be the diagonal map.}\\ & x& ⟼& (x,x)\end{array}$

Then

$\begin{array}{ccc}\begin{array}{ccccc}X& \stackrel{\nu }{⟶}& PX& \stackrel{\rho }{⟶}& X\times X\\ & & \gamma & ⟼& (\gamma \left(0\right),\gamma \left(1\right))\\ x& ⟼& C_x\end{array}& \text{where}& \begin{array}{cccc}\multicolumn{4}{c}{PX=\text{Map}([0,1],X)}\\ C_x:& [0,1]& ⟶& X\\ & t& ⟼& x\end{array}\end{array}$

with $\nu$ a homotopy equivalence and $\rho$ a fibration.

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