## Fibrations

Last update: 9 September 2012

## Introduction

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

Let $\iota :\phantom{\rule{0.2em}{0ex}}A⟶X$ be a morphism and let $Y\in \text{Top}$.

The morphism $\iota :\phantom{\rule{0.2em}{0ex}}A⟶X$ has the homotopy extension property with respect to $Y$ if $\iota$ satisfies:

$\begin{array}{c}\text{if}\phantom{\rule{1em}{0ex}}f:\phantom{\rule{0.2em}{0ex}}X⟶Y\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}h:\phantom{\rule{0.2em}{0ex}}A⟶\text{Hom}\phantom{\rule{0.2em}{0ex}}\left(\left[0,1\right],Y\right)\phantom{\rule{1em}{0ex}}\text{are morphisms such that}\\ \text{if}\phantom{\rule{1em}{0ex}}a\in A\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}\left(h\left(a\right)\right)\left(0\right)=f\left(\iota \left(a\right)\right)\\ \text{then there exists}\phantom{\rule{1em}{0ex}}\stackrel{\sim }{h}:\phantom{\rule{0.2em}{0ex}}X⟶\text{Hom}\phantom{\rule{0.2em}{0ex}}\left(\left[0,1\right],Y\right)\phantom{\rule{1em}{0ex}}\text{such that}\\ \text{if}\phantom{\rule{1em}{0ex}}x\in X\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}\left(\stackrel{\sim }{h}\left(x\right)\right)\left(0\right)=f\left(x\right)\text{.}\end{array}$

A cofibration is a morphism $\iota :\phantom{\rule{0.2em}{0ex}}A⟶X$ such that if mat yTop then $\iota :\phantom{\rule{0.2em}{0ex}}A⟶X$ has the homotopy extension property with respect to $Y$.

The cofibre of a cofibration $\iota :\phantom{\rule{0.2em}{0ex}}A⟶X$ is $\frac{X}{\iota \left(X\right)}$ and

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

is the cofibration sequence of $\iota :\phantom{\rule{0.2em}{0ex}}A⟶X$.

Let $f:\phantom{\rule{0.2em}{0ex}}X⟶Y$ be a morphism.

The mapping cone of $f$ is the pushout $\text{Con}\phantom{\rule{0.2em}{0ex}}\left(f\right)$,

and ${\iota }_{0}:\phantom{\rule{0.2em}{0ex}}X⟶CX$ is given by ${\iota }_{0}\left(x\right)=\left(x,0\right)$.

The mapping cylinder of $f$, or homotopy cofibre of $f$, is the pushout $\text{Cyl}\phantom{\rule{0.2em}{0ex}}\left(f\right)$.

$\begin{array}{ccc}\begin{array}{ccc}X& \stackrel{f}{⟶}& Y\\ {\iota }_{0}↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}\\ X×\left[0,1\right]& ⟶& \text{Cyl}\phantom{\rule{0.2em}{0ex}}\left(f\right)\end{array}& ,\phantom{\rule{1em}{0ex}}\text{where}& \begin{array}{cccc}{\iota }_{0}:& X& ⟶& X×\left[0,1\right]\\ & x& ⟼& \left(x,0\right)\end{array}\end{array}$

Let $f:\phantom{\rule{0.2em}{0ex}}X⟶Y$ be a morphism.

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

$\begin{array}{cc}\begin{array}{cccccc}f:& X& \stackrel{j}{⟶}& \text{Cyl}\phantom{\rule{0.2em}{0ex}}\left(f\right)& \stackrel{\tau }{⟶}& Y\\ & x& ⟼& \left(x,1\right)\\ & & & y& ⟼& y\\ & & & \left(x,t\right)& ⟼& f\left(x\right)\end{array}& \text{and}\phantom{\rule{0.2em}{0ex}}\tau \phantom{\rule{0.2em}{0ex}}\text{is a homotopy equivalence.}\end{array}$

2. The morphism $j:\phantom{\rule{0.2em}{0ex}}X⟶\text{Cyl}\phantom{\rule{0.2em}{0ex}}\left(f\right)$ is a cofibration with cofibration sequence

$\begin{array}{ccccc}X& \stackrel{j}{⟶}& \text{Cyl}\phantom{\rule{0.2em}{0ex}}\left(f\right)& ⟶& \text{Con}\phantom{\rule{0.2em}{0ex}}\left(f\right)\end{array}$

Let $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ be a morphism and let $Y\in \text{Top}$.

The morphism $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ has the homotopy lifting property with respect to $Y$ if $p$ satisfies:

$\begin{array}{c}\text{if}\phantom{\rule{1em}{0ex}}f:Y⟶E\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}h:\phantom{\rule{0.2em}{0ex}}Y×\left[0,1\right]⟶B\phantom{\rule{1em}{0ex}}\text{are morphisms such that}\\ \text{if}\phantom{\rule{1em}{0ex}}y\in Y\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}h\left(y,0\right)=p\left(f\left(y\right)\right)\\ \text{then there exists}\phantom{\rule{1em}{0ex}}\stackrel{\sim }{h}:\phantom{\rule{0.2em}{0ex}}Y×\left[0,1\right]⟶E\phantom{\rule{1em}{0ex}}\text{such that}\\ \text{if}\phantom{\rule{1em}{0ex}}\left(y,t\right)\in Y×\left[0,1\right]\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}h\left(y,t\right)=p\left(\stackrel{\sim }{h}\left(y,t\right)\right)\end{array}$

A Hurewicz fibration is a morphism $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ such that if $Y\in \text{Top}$ then $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ satisfies the homotopy lifting property with respect to $Y$.

A fibration, or Serre fibration is a morphism $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ such that,

if $Y\in CW$ then $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ satisfies the homotopy lifting property with respect to $Y\text{.}$

The fibre of a fibration $p:\phantom{\rule{0.2em}{0ex}}E⟶B$ 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 $p:\phantom{\rule{0.2em}{0ex}}E⟶B$.

Let $f:\phantom{\rule{0.2em}{0ex}}X⟶Y$ be a morphism.

The homotopy fibre of $f$ is the pullback ${F}_{f}$,

$\begin{array}{ccc}\begin{array}{ccc}{F}_{f}& ⟶& X\\ ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}f\\ {\text{Map}}_{✶}\phantom{\rule{0.2em}{0ex}}\left(\left[0,1\right],y\right)& \underset{{p}_{1}}{⟶}& Y\end{array}& \text{where}& \begin{array}{cccc}{p}_{1}:& {\text{Map}}_{✶}\phantom{\rule{0.2em}{0ex}}\left(\left[0,1\right],y\right)& ⟶& Y\\ & \beta & ⟼& \beta \left(1\right)\end{array}\end{array}$

The mapping path space of $f$ is the pullback ${N}_{f}$

$\begin{array}{ccc}\begin{array}{ccc}{N}_{f}& ⟶& \text{Map}\phantom{\rule{0.2em}{0ex}}\left(\left[0,1\right],y\right)\\ ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}{p}_{1}\\ X& \underset{f}{⟶}& Y\end{array}& \text{where}& \begin{array}{cccc}{p}_{1}:& \text{Map}\phantom{\rule{0.2em}{0ex}}\left(\left[0,1\right],y\right)& ⟶& Y\\ & \beta & ⟼& \beta \left(1\right)\end{array}\end{array}$

Let $f:\phantom{\rule{0.2em}{0ex}}X⟶Y$ be a morphism.

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

$\begin{array}{cc}\begin{array}{cccccc}f:& X& \stackrel{\nu }{⟶}& {N}_{f}& \stackrel{\rho }{⟶}& Y\\ & x& ⟼& \left(x,{C}_{f\left(x\right)}\right)\\ & & & \left(x,\beta \right)& ⟼& \beta \left(0\right)\end{array}& \text{and}\phantom{\rule{0.2em}{0ex}}\nu \phantom{\rule{0.2em}{0ex}}\text{is a homotopy equivalence,}\end{array}$

where, something, ${C}_{y}:\phantom{\rule{0.2em}{0ex}}\left[0,1\right]⟶Y$ is given by ${C}_{y}\left(t\right)=y$.

2. The morphism $\rho :\phantom{\rule{0.2em}{0ex}}{N}_{f}⟶Y$ 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 ${\pi }_{1}\left(B,{b}_{0}\right)$ 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. ${\pi }_{1}\left(E,{e}_{0}\right)=0$.

(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 $p=h\circ {p}^{\prime }$

$\begin{array}{ccc}E& \stackrel{h}{⟶}& {E}^{\prime }\\ p↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}& & ↓\phantom{\rule[-1.5ex]{0ex}{1.5ex}}{p}^{\prime }\\ B& \underset{{\text{id}}_{B}}{⟶}& B\end{array}$

[Benson II, Theorem 1.16.13]. The map

$\begin{array}{ccc}\left\{\genfrac{}{}{0}{}{\text{conjugacy classes of}}{\text{subgroups}\phantom{\rule{0.2em}{0ex}}H\subseteq {\pi }_{1}\left(B,{b}_{0}\right)}\right\}& ⟶& \left\{\genfrac{}{}{0}{}{\text{isomorphism classes of}}{\text{covers}\phantom{\rule{0.2em}{0ex}}{E}^{\prime }\stackrel{{p}^{\prime }}{⟶}B}\right\}\\ H& ⟼& \left(E,{e}_{0}\right)/H\end{array}$

## Nerves and classifying spaces

Let $𝒞$ be a category.

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

$𝒩{𝒞}_{n}=\left\{\text{sequences}\phantom{\rule{1em}{0ex}}{M}_{1}\stackrel{{f}_{1}}{⟶}{M}_{2}\stackrel{{f}_{2}}{⟶}\dots \stackrel{{f}_{n}}{⟶}{M}_{n+1}\phantom{\rule{1em}{0ex}}\text{of morphisms in}\phantom{\rule{0.2em}{0ex}}𝒞\right\}$

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

$\text{objects:}\phantom{\rule{0.2em}{0ex}}g\in G\phantom{\rule{1em}{0ex}}\text{morphism:}\phantom{\rule{0.2em}{0ex}}\left\{g\stackrel{h}{⟶}hg\right\}=\text{Hom}\phantom{\rule{0.2em}{0ex}}\left(g,hg\right)$

The category of $G$ has

$\text{one object:}\phantom{\rule{0.2em}{0ex}}\cdot ,\phantom{\rule{1em}{0ex}}\text{and morphisms}\phantom{\rule{1em}{0ex}}\text{Hom}\phantom{\rule{0.2em}{0ex}}\left(\cdot ,\cdot \right)=G$

HW: Show that

$\mid 𝒩𝒞\left(G\right)\mid =EG,\phantom{\rule{1em}{0ex}}𝒩G=𝒩𝒞\left(G\right){/}_{G},\phantom{\rule{1em}{0ex}}\mid 𝒩𝒞\left(G\right){/}_{G}\mid =\mathrm{Bscr;}G<\text{.}>$

### 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 $n\ne 6,7,$

$0⟶ℤ/2ℤ⟶{\stackrel{\sim }{A}}_{n}⟶{A}_{n}⟶1$

is a universal central extension of ${A}_{n}$ (obtained by restricting the central extension

$0⟶ℤ/2ℤ⟶{\text{Spin}}_{n-1}\phantom{\rule{0.2em}{0ex}}\left(ℝ\right)⟶S{O}_{n-1}⟶1.\right)$

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

$r\in R,\phantom{\rule{0.2em}{0ex}}1\le i,j\le n,$

with relations

$\begin{array}{c}{x}_{ij}\left({r}_{1}\right){x}_{ih}\left({r}_{2}\right)={x}_{ij}\left({r}_{1}+{r}_{2}\right)\\ \left[{x}_{ij}\left({r}_{1}\right),{x}_{jk}\left({r}_{2}\right)\right]={x}_{ik}\left({r}_{1}{r}_{2}\right),\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}i\ne k\\ \left[{x}_{ij}\left({r}_{1}\right),{x}_{kl}\left({r}_{2}\right)\right]=1,\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}j\ne k\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}i\ne l\text{.}\end{array}$

Then, for $n\ge 5$,

$1⟶{K}_{2}\left(u,R\right)⟶S{t}_{n}\left(R\right)⟶{E}_{n}\left(R\right)⟶1$

is a universal central extension of ${E}_{n}\left(R\right)$.

The Milnor $K$–group is

${K}_{2}\left(R\right)=\underset{⟶}{\text{lim}}\phantom{\rule{0.2em}{0ex}}{K}_{2}\left(n,R\right)\text{.}$

## 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 ${\pi }_{1}\left({S}^{1},{s}_{0}\right)=ℤ$
2. ${S}^{n}⟶ℝ{P}^{n}$ is a universal cover with ${\pi }_{1}\left({S}^{1}×{S}^{1},{x}_{0}\right)=ℤ/2$
3. ${R}^{2}⟶{S}^{1}×{S}^{1}$ is a universal cover with ${\pi }_{1}\left({S}^{1}×{S}^{1},{x}_{0}\right)=ℤ×ℤ$
4. ${S}^{n}⟶ℝ{P}^{n}$ is a principal $ℤ/2ℤ$–bundle. $\left(ℤ/2ℤ=O\left(1\right)={O}_{1}\left(ℝ\right)\right)$
5. ${S}^{\infty }⟶ℝ{P}^{\infty }$ is a universal principal $ℤ/2ℤ$–bundle.
6. $ℝ⟶{S}^{1}$ is a universal principal $ℤ$–bundle.
7. $0⟶ℤ\stackrel{2\pi i}{⟶}ℂ\stackrel{\text{exp}}{⟶}{ℂ}^{×}⟶1$ is a universal principal $ℤ$–bundle.
8. ${S}^{2n+1}⟶ℂ{P}^{n}$ is a principal ${S}^{1}$–bundle. $\left({S}^{1}=U\left(1\right)\right)$
9. ${S}^{\infty }⟶ℂ{P}^{\infty }$ is a universal principal ${S}^{1}$–bundle.
10. Let ${V}_{k}\left({ℝ}^{n}\right)=\left\{\text{orhonormal}\phantom{\rule{0.2em}{0ex}}k\text{–frames in}\phantom{\rule{0.2em}{0ex}}{ℝ}^{n}\right\}$ be the Stiefel manifold, and

${G}_{k}\left({ℝ}^{n}\right)=\left\{k\phantom{\rule{0.2em}{0ex}}\text{dimensional subspaces in}\phantom{\rule{0.2em}{0ex}}{ℝ}^{n}\right\}\phantom{\rule{1em}{0ex}}\text{the}$

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

${V}_{k}\left({ℝ}^{n}\right)⟶G{r}_{k}\left({ℝ}^{n}\right)\phantom{\rule{1em}{0ex}}\text{is a principal}\phantom{\rule{1em}{0ex}}O\left(k\right)\text{–bundle.}$

11. ${V}_{k}\left({ℝ}^{\infty }\right)⟶{G}_{k}\left({ℝ}^{\infty }\right)\phantom{\rule{1em}{0ex}}\text{is a universal principal}\phantom{\rule{1em}{0ex}}O\left(k\right)\text{–bundle.}$
12. Spheres:

$\begin{array}{c}{S}^{n-1}\phantom{\rule{1em}{0ex}}\text{is a sphere in}\phantom{\rule{1em}{0ex}}{ℝ}^{n},\\ {S}^{2n-1}\phantom{\rule{1em}{0ex}}\text{is a sphere in}\phantom{\rule{1em}{0ex}}{ℂ}^{n},\\ {S}^{4n-1}\phantom{\rule{1em}{0ex}}\text{is a sphere in}\phantom{\rule{1em}{0ex}}{ℍ}^{n},\end{array}$

and

$\begin{array}{c}SO\left(n-1\right)⟶SO\left(n\right)⟶{S}^{n-1}\phantom{\rule{1em}{0ex}}\text{is a principal}\phantom{\rule{1em}{0ex}}SO\left(n-1\right)\phantom{\rule{1em}{0ex}}\text{bundle,}\\ \begin{array}{c}U\left(n-1\right)⟶U\left(n\right)⟶{S}^{2n-1}\phantom{\rule{1em}{0ex}}\text{is a principal}\phantom{\rule{1em}{0ex}}U\left(n-1\right)\phantom{\rule{1em}{0ex}}\text{bundle,}\\ \begin{array}{c}SU\left(n-1\right)⟶SU\left(n\right)⟶{S}^{2n-1}\phantom{\rule{1em}{0ex}}\text{is a principal}\phantom{\rule{1em}{0ex}}SU\left(n-1\right)\phantom{\rule{1em}{0ex}}\text{bundle,}\\ \begin{array}{c}Sp\left(n-1\right)⟶Sp\left(n\right)⟶{S}^{4n-1}\phantom{\rule{1em}{0ex}}\text{is a principal}\phantom{\rule{1em}{0ex}}Sp\left(n-1\right)\phantom{\rule{1em}{0ex}}\text{bundle,}\end{array}\end{array}\end{array}\end{array}$

where in each case the map is given by

$A⟼{A}_{{e}_{n}}\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}{e}_{n}=\left(0,0,\dots ,0,1\right)\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{1em}{0ex}}{ℝ}^{n},\phantom{\rule{0.2em}{0ex}}{ℂ}^{n}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}{ℍ}^{n}\text{.}$

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

$ℝ{P}^{1}={S}^{1},\phantom{\rule{1em}{0ex}}ℂ{P}^{1}={S}^{2},\phantom{\rule{1em}{0ex}}ℍ{P}^{1}={S}^{4},\phantom{\rule{1em}{0ex}}𝕆{P}^{1}={S}^{8}$

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

$\begin{array}{ccc}\left({S}^{1}⟶{S}^{1}=ℝ{P}^{1}\right)& =& \left({U}_{1}\left(ℝ\right)⟶SO\left(2\right)⟶ℝ{P}^{1}\right)\\ \left({S}^{3}⟶{S}^{2}=ℂ{P}^{1}\right)& =& \left({U}_{1}\left(ℂ\right)⟶SU\left(2\right)⟶ℂ{P}^{1}\right)\\ \left({S}^{7}⟶{S}^{4}=ℍ{P}^{1}\right)& =& \left({U}_{1}\left(ℍ\right)⟶S{U}_{2}\left(ℍ\right)⟶ℍ{P}^{1}\right)\\ \left({S}^{15}⟶{S}^{8}=𝕆{P}^{1}\right)& =& \left({U}_{1}\left(𝕆\right)⟶S{U}_{2}\left(𝕆\right)⟶𝕆{P}^{1}\right)\end{array}$

where the map is $v⟼\text{span}\phantom{\rule{0.2em}{0ex}}\left(v\right)$ in each case.

14. Projective Spaces

O(n-1) SO(n) Pn-1 U(n-1) SU(n) Pn-1

are principal bundles.

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

${G}_{k}\left({ℝ}^{n}\right)=\left\{k\phantom{\rule{0.2em}{0ex}}\text{dimensional subspaces of}\phantom{\rule{0.2em}{0ex}}{ℝ}^{n}\right\}\text{.}$

The sequences

$\begin{array}{cccc}O\left(k\right)×O\left(n-k\right)& ⟶& O\left(n\right)& ⟶{G}_{k}\left({ℝ}^{n}\right)\\ U\left(k\right)×U\left(n-k\right)& ⟶& U\left(n\right)& ⟶{G}_{k}\left({ℂ}^{n}\right)\\ {S}_{p}\left(k\right)×{S}_{p}\left(n-k\right)& ⟶& {S}_{p}\left(n\right)& ⟶{G}_{k}\left({ℍ}^{n}\right)\end{array}$

are principal bundles.

16. The Steifel manifold is

${V}_{k}\left({ℝ}^{n}\right)=\left\{\text{orthonormal}\phantom{\rule{0.2em}{0ex}}k\text{–frames in}\phantom{\rule{0.2em}{0ex}}{ℝ}^{n}\right\}$.

The sequence

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

is a principal bundle.

17. $\begin{array}{ccc}{S}^{1}& ⟶& {S}^{1}\\ z& ⟼& {z}^{n}\end{array}$ is a principal $ℤ/nℤ$ 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

${a}_{1},\dots ,{a}_{g},\phantom{\rule{1em}{0ex}}{b}_{1},\dots ,{b}_{g}$

with relations

$\left[{a}_{1},{b}_{1}\right]\left[{a}_{2},{b}_{2}\right]\dots \left[{a}_{g},{b}_{g}\right]=1\text{.}$

Then

$ℋ⟶S$ is a universal principal bundle for ${\pi }_{1}\left(S\right)$,

where $ℋ$ 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

$G}{K}⟶\Gamma \setminus G}{K}$ is a universal principal $\Gamma$–bundle.

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

For $N\ge 3$,

$\Gamma \left(N\right)=\left\{\gamma \in S{L}_{n}\left(ℤ\right)\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\gamma =1\phantom{\rule{0.2em}{0ex}}\text{mod}\phantom{\rule{0.2em}{0ex}}N\right\}$

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

$S{L}_{n}\left(ℝ\right)}{S{O}_{n}\left(ℝ\right)}⟶\Gamma \left(N\right)\setminus S{L}_{n}\left(ℝ\right)}{S{O}_{n}\left(ℝ\right)}$ 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, $\phantom{\rule{2em}{0ex}}$], [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

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

Then

$\begin{array}{ccc}\begin{array}{ccccc}X& \stackrel{\nu }{⟶}& PX& \stackrel{\rho }{⟶}& X×X\\ & & \gamma & ⟼& \left(\gamma \left(0\right),\gamma \left(1\right)\right)\\ x& ⟼& {C}_{x}\end{array}& \text{where}& \begin{array}{cccc}\multicolumn{4}{c}{PX=\text{Map}\phantom{\rule{0.2em}{0ex}}\left(\left[0,1\right],X\right)}\\ {C}_{x}:& \left[0,1\right]& ⟶& X\\ & t& ⟼& x\end{array}\end{array}$

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