User:Moritz Metzger/covering

A covering of a topological space $X$ is a continuous map $\pi :E\rightarrow X$ with special properties.

Definition

Let $X$ be a topological space. A covering of $X$ is a continuous map

$\pi :E\rightarrow X$

s.t. there exists a discrete space $D$ and for every $x\in X$ an open neighborhood $U\subset X$, s.t. $\pi ^{-1}(U)=\displaystyle \bigsqcup _{d\in D}V_{d}$ and $\pi |_{V_{d}}:V_{d}\rightarrow U$ is a homeomorphism for every $d\in D$.

Intuitively, a covering locally project a "stack of pancakes" above an open neighborhood $U$ onto $U$

Often, the notion of a covering is used for the covering space $E$ as well as for the map $\pi :E\rightarrow X$. The open sets $V_{d}$ are called sheets, which are uniquely determined if $U$ is connected.$\,^{p.56}$ For a $x\in X$ the discrete subset $\pi ^{-1}(x)$ is called the fiber of $x$. The degree of a covering is the cardinality of the space $D$. If $E$ is path-connected, then the covering $\pi :E\rightarrow X$ is denoted as a path-connected covering.

Examples

• For every topological space $X$ there exists the covering $\pi :X\rightarrow X$ with $\pi (x)=x$, which is denoted as the trivial covering of $X.$

The space $Y=[0,1]\times \mathbb {R}$ is the covering space of $X=[0,1]\times S^{1}$. The disjoint open sets $S_{i}$ are mapped homeomorphically onto $U$. The fiber of $x$ consists of the points $y_{i}$.

• The map $r\colon \mathbb {R} \to S^{1}$ with $r(t)=(\cos(2\pi t),\sin(2\pi t))$ is a covering of the unit circle $S^{1}$. For an open neighborhood $U\subset X$ of an $x\in S^{1}$, which has positiv $\cos(2\pi t)$-value, one has: $r^{-1}(U)=\displaystyle \bigsqcup _{n\in \mathbb {Z} }(n-{\frac {1}{4}},n+{\frac {1}{4}})$.
• Another covering of the unit circle is the map $q\colon S^{1}\to S^{1}$ with $q(z)=z^{n}$ for some $n\in \mathbb {N}$. For an open neighborhood $U\subset X$ of an $x\in S^{1}$, one has: $q^{-1}(U)=\displaystyle \bigsqcup _{i=1}^{n}U$.
• A map which is a local homeomorphism but not a covering of the unit circle is $p\colon \mathbb {R_{+}} \to S^{1}$ with $p(t)=(\cos(2\pi t),\sin(2\pi t))$. There is a sheet of an open neighborhood of $(1,0)$, which is not mapped homeomorphically onto $U$.

Properties

Local homeomorphism

Since a covering $\pi :E\rightarrow X$ maps each of the disjoint open sets of $\pi ^{-1}(U)$ homeomorphically onto $U$ it is a local homeomorphism, i.e. $\pi$ is a continuous map and for every $e\in E$ there exists an open neighborhood $V\subset E$ of $e$, s.t. $\pi |_{V}:V\rightarrow \pi (V)$ is a homeomorphism.

It follows that the covering space $E$ and the base space $X$ locally share the same properties.

• If $X$ is a connected and non-orientable manifold, then there is a covering $\pi :{\tilde {X}}\rightarrow X$ of degree $2$, whereby ${\tilde {X}}$ is a connected and orientable manifold.$\,^{p.234}$
• If $X$ is a connected Lie group, then there is a covering $\pi :{\tilde {X}}\rightarrow X$ which is also a Lie group homomorphism and ${\tilde {X}}:=\{\gamma :\gamma {\text{ is a path in X with }}\gamma (0)={\boldsymbol {1_{X}}}{\text{ modulo homotopy with fixed ends}}\}$ is a Lie group. $^{p.174}$
• If $X$ is a graph, then it follows for a covering $\pi :E\rightarrow X$ that $E$ is also a graph. $^{p.85}$
• If $X$ is a connected manifold, then there is a covering $\pi :{\tilde {X}}\rightarrow X$, whereby ${\tilde {X}}$ is a connected and simply connected manifold. $^{p.32}$
• If $X$ is a connected Riemann surface, then there is a covering $\pi :{\tilde {X}}\rightarrow X$ which is also a holomorphic map $^{p.22}$ and ${\tilde {X}}$ is a connected and simply connected Riemann surface. $^{p.32}$

Factorisation

Let $p,q$ and $r$ be continuous maps, s.t. the diagram

commutes.

• If $p$ and $q$ are coverings, so is $r$. $^{p.485}$
• If $p$ and $r$ are coverings, so is $q$. $^{p.485}$

Product of coverings

Let $X$ and $X'$ be topological spaces and $p:E\rightarrow X$ and $p':E'\rightarrow X'$ be coverings, then $p\times p':E\times E'\rightarrow X\times X'$ with $(p\times p')(e,e')=(p(e),p'(e'))$ is a covering. $^{p.339}$

Equivalence of coverings

Let $X$ be a topological space and $p:E\rightarrow X$ and $p':E'\rightarrow X$ be coverings. Both coverings are called equivalent, if there exists a homeomorphism $h:E\rightarrow E'$, s.t. the diagram

commutes. If such a homeomorphism exists, then one calls the covering spaces $E$ and $E'$ isomorphic.

Lifting property

An important property of the covering is, that it satisfies the lifting property, i.e.:

Let $I$ be the unit interval and $p:E\rightarrow X$ be a covering. Let $F:Y\times I\rightarrow X$ be a continuous map and ${\tilde {F}}_{0}:Y\times \{0\}\rightarrow E$ be a lift of $F|_{Y\times \{0\}}$, i.e. a continuous map such that $p\circ {\tilde {F}}_{0}=F|_{Y\times \{0\}}$. Then there is a uniquely determined, continuous map ${\tilde {F}}:Y\times I\rightarrow E$, which is a lift of $F$, i.e. $p\circ {\tilde {F}}=F$. $^{p.60}$

If $X$ is a path-connected space, then for $Y=\{0\}$ it follows that the map ${\tilde {F}}$ is a lift of a path in $X$ and for $Y=I$ it is a lift of a homotopy of paths in $X$.

Because of that property one can show, that the fundamental group $\pi _{1}(S^{1})$ of the unit circle is an infinite cyclic group, which is generated by the homotopy classes of the loop $\gamma :I\rightarrow S^{1}$ with $\gamma (t)=(\cos(2\pi t),\sin(2\pi t))$. $^{p.29}$

Let $X$ be a path-connected space and $p:E\rightarrow X$ be a connected covering. Let $x,y\in X$ be any two points, which are connected by a path $\gamma$, i.e. $\gamma (0)=x$ and $\gamma (1)=y$. Let ${\tilde {\gamma }}$ be the unique lift of $\gamma$, then the map

$L_{\gamma }:p^{-1}(x)\rightarrow p^{-1}(y)$ with $L_{\gamma }({\tilde {\gamma }}(0))={\tilde {\gamma }}(1)$

is bijective. $^{p.69}$

If $X$ is a path-connected space and $p:E\rightarrow X$ a connected covering, then the induced group homomorphism

$p_{\#}:\pi _{1}(E)\rightarrow \pi _{1}(X)$ with $p_{\#}([\gamma ])=[p\circ \gamma ]$,

is injective and the subgroup $p_{\#}(\pi _{1}(E))$ of $\pi _{1}(X)$ consists of the homotopy classes of loops in $X$, whose lifts are loops in $E$. $^{p.61}$

Branched covering

Definitions

Holomorphic maps between Riemann surfaces

Let $X$ and $Y$ be Riemann surfaces, i.e. one dimensional complex manifolds, and let $f:X\rightarrow Y$ be a continuous map. $f$ is holomorphic in a point $x\in X$, if for any charts $\phi _{x}:U_{1}\rightarrow V_{1}$ of $x$ and $\phi _{f(x)}:U_{2}\rightarrow V_{2}$ of $f(x)$, with $\phi _{x}(U_{1})\subset U_{2}$, the map $\phi _{f(x)}\circ f\circ \phi _{x}^{-1}:\mathbb {C} \rightarrow \mathbb {C}$ is holomorphic.

If $f$ is for all $x\in X$ holomorphic, we say $f$ is holomorphic.

The map $F=\phi _{f(x)}\circ f\circ \phi _{x}^{-1}$ is called the local expression of $f$ in $x\in X$.

If $f:X\rightarrow Y$ is a non-constant, holomorphic map between compact Riemann surfaces, then $f$ is surjective $^{p.11}$ and an open map $^{p.11}$, i.e. for every open set $U\subset X$ the image $f(U)\subset Y$ is also open.

Ramification point and branch point

Let $f:X\rightarrow Y$ be a non-constant, holomorphic map between compact Riemann surfaces. For every $x\in X$ there exist charts for $x$ and $f(x)$ and there exists a uniquely determined $k_{x}\in \mathbb {N_{>0}}$, s.t. the local expression $F$ of $f$ in $x$ is of the form $z\mapsto z^{k_{x}}$. $^{p.10}$ The number $k_{x}$ is called the ramification index of $f$ in $x$ and the point $x\in X$ is called a ramification point if $k_{x}\geq 2$. If $k_{x}=1$ for an $x\in X$, then $x$ is unramified. The image point $y=f(x)\in Y$ of a ramification point is called a branch point.

Degree of a holomorphic map

Let $f:X\rightarrow Y$ be a non-constant, holomorphic map between compact Riemann surfaces. The degree $deg(f)$ of $f$ is the cardinality of the fiber of an unramified point $y=f(x)\in Y$, i.e. $deg(f):=|f^{-1}(y)|$.

This number is well-defined, since for every $y\in Y$ the fiber $f^{-1}(y)$ is discrete $^{p.20}$ and for any two unramified points $y_{1},y_{2}\in Y$, it is: $|f^{-1}(y_{1})|=|f^{-1}(y_{2})|.$ $^{p.29}$

It can be calculated by:

$\sum _{x\in f^{-1}(y)}k_{x}=deg(f)$ $^{p.29}$

Branched covering

Definition

A continuous map $f:X\rightarrow Y$ is called a branched covering, if there exists a closed set with dense complement $E\subset Y$, s.t. $f_{|X\smallsetminus f^{-1}(E)}:X\smallsetminus f^{-1}(E)\rightarrow Y\smallsetminus E$ is a covering.

Examples

• Let $n\in \mathbb {N}$ and $n\geq 2$, then $f:\mathbb {C} \rightarrow \mathbb {C}$ with $f(z)=z^{n}$ is branched covering of degree $n$, whereby $z=0$ is a branch point.
• Every non-constant, holomorphic map between compact Riemann surfaces $f:X\rightarrow Y$ of degree $d$ is a branched covering of degree $d$.

Universal covering

Let $p:{\tilde {X}}\rightarrow X$ be a simply connected covering and $\beta :E\rightarrow X$ be a covering, then there exists a uniquely determined covering $\alpha :{\tilde {X}}\rightarrow E$, s.t. the diagram

commutes. $^{p.486}$

Definition

Let $p:{\tilde {X}}\rightarrow X$ be a simply connected covering. If $\beta :E\rightarrow X$ is another simply connected covering, then there exists a uniquely determined homeomorphism $\alpha :{\tilde {X}}\rightarrow E$, s.t. the diagram

commutes. $^{p.482}$

This means that $p$ is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space $X$.

Existence

A universal covering does not always exists, but the following properties guarantee the existence:

Let $X$ be a connected, locally simply connected, then there exists a universal covering $p:{\tilde {X}}\rightarrow X$.

${\tilde {X}}$ is defined as ${\tilde {X}}:=\{\gamma :\gamma {\text{ is a path in }}X{\text{ with }}\gamma (0)=x_{0}\}/{\text{ homotopy with fixed ends}}$ and $p:{\tilde {X}}\rightarrow X$ by $p([\gamma ]):=\gamma (1)$. $^{p.64}$

The topology on ${\tilde {X}}$ is constructed as follows: Let $\gamma :I\rightarrow X$ be a path with $\gamma (0)=x_{0}$. Let $U$ be a simply connected neighborhood of the endpoint $x=\gamma (1)$, then for every $y\in U$ the paths $\sigma _{y}$ inside $U$ from $x$ to $y$ are uniquely determined up to homotopy. Now consider ${\tilde {U}}:=\{\gamma .\sigma _{y}:y\in U\}/{\text{ homotopy with fixed ends}}$, then $p_{|{\tilde {U}}}:{\tilde {U}}\rightarrow U$ with $p([\gamma .\sigma _{y}])=\gamma .\sigma _{y}(1)=y$ is a bijection and ${\tilde {U}}$ can be equipped with the final topology of $p_{|{\tilde {U}}}$.

The fundamental group $\pi _{1}(X,x_{0})=\Gamma$ acts freely through $([\gamma ],[{\tilde {x}}])\mapsto [\gamma .{\tilde {x}}]$ on ${\tilde {X}}$ and $\psi :\Gamma \backslash {\tilde {X}}\rightarrow X$ with $\psi ([\Gamma {\tilde {x}}])={\tilde {x}}(1)$ is a homeomorphism, i.e. $\Gamma \backslash {\tilde {X}}\cong X$.

Examples

• $r\colon \mathbb {R} \to S^{1}$ with $r(t)=(\cos(2\pi t),\sin(2\pi t))$ is the universal covering of the unit circle $S^{1}$.
• $p\colon S^{n}\to \mathbb {R} P^{n}\cong \{+1,-1\}\backslash S^{n}$ with $p(x)=[x]$ is the universal covering of the projective space $\mathbb {R} P^{n}$ for $n>1$.
• $q\colon SU(n)\ltimes \mathbb {R} \to U(n)$ with $q(A,t)={\begin{bmatrix}\exp(2\pi it)&0\\0&I_{n-1}\end{bmatrix}}A$ is the universal covering of the unitary group $U(n)$.
• Since $SU(2)\cong S^{3}$, it follows that the quotient map $f:SU(2)\rightarrow \mathbb {Z_{2}} \backslash SU(2)\cong SO(3)$ is the universal covering of the $SO(3)$.

  

  The Hawaiian earring. Only the ten largest circles are shown.

• A topological space, which has no universal covering is the Hawaiian earring:

$X=\bigcup _{n\in \mathbb {N} }\left\{(x_{1},x_{2})\in \mathbb {R} ^{2}:{\Bigl (}x_{1}-{\frac {1}{n}}{\Bigr )}^{2}+x_{2}^{2}={\frac {1}{n^{2}}}\right\}$

One can show, that no neighborhood of the origin $(0,0)$ is simply connected. $^{p.487\,Example\,1}$

Deck transformation

Definition

Let $p:E\rightarrow X$ be a covering. A deck transformation is a homeomorphism $d:E\rightarrow E$, s.t. the diagram of continuous maps

commutes. Together with the composition of maps, the set of deck transformation forms a group $Deck(p)$, which is the same as $Aut(p)$.

Examples

• Let $q\colon S^{1}\to S^{1}$ be the covering $q(z)=z^{n}$ for some $n\in \mathbb {N}$, then the map $d:S^{1}\rightarrow S^{1}:z\mapsto z\,e^{2\pi i/n}$ is a deck transformation and $Deck(q)\cong \mathbb {Z} /\mathbb {nZ}$.
• Let $r\colon \mathbb {R} \to S^{1}$ be the covering $r(t)=(\cos(2\pi t),\sin(2\pi t))$, then the map $d_{k}:\mathbb {R} \rightarrow \mathbb {R} :t\mapsto t+k$ with $k\in \mathbb {Z}$ is a deck transformation and $Deck(r)\cong \mathbb {Z}$.

Properties

Let $X$ be a path-connected space and $p:E\rightarrow X$ be a connected covering. Since a deck transformation $d:E\rightarrow E$ is bijective, it permutes the elements of a fiber $p^{-1}(x)$ with $x\in X$ and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber. $^{p.70}$Because of this property every deck transformation defines a group action on $E$, i.e. let $U\subset X$ be an open neighborhood of a $x\in X$ and ${\tilde {U}}\subset E$ an open neighborhood of an $e\in p^{-1}(x)$, then $Deck(p)\times E\rightarrow E:(d,{\tilde {U}})\mapsto d({\tilde {U}})$ is a group action.

Normal coverings

Definition

A covering $p:E\rightarrow X$ is called normal, if $Deck(p)\backslash X\cong E$. This means, that for every $x\in X$ and any two $e_{0},e_{1}\in p^{-1}(x)$ there exists a deck transformation $d:E\rightarrow E$, s.t. $d(e_{0})=e_{1}$.

Properties

Let $X$ be a path-connected space and $p:E\rightarrow X$ be a connected covering. Let $H=p_{\#}(\pi _{1}(E))$ be a subgroup of $\pi _{1}(X)$, then $p$ is a normal covering iff $H$ is a normal subgroup of $\pi _{1}(X)$. $^{p.71}$

If $p:E\rightarrow X$ is a normal covering and $H=p_{\#}(\pi _{1}(E))$, then $Deck(p)\cong \pi _{1}(X)/H$. $^{p.71}$

If $p:E\rightarrow X$ is a path-connected covering and $H=p_{\#}(\pi _{1}(E))$, then $Deck(p)\cong N(H)/H$, whereby $N(H)$ is the normaliser of $H$. $^{p.71}$

Let $E$ be a topological space. A group $\Gamma$ acts discontinuously on $E$, if every $e\in E$ has an open neighborhood $V\subset E$ with $V\neq \emptyset$, such that for every $\gamma \in \Gamma$ with $\gamma V\cap V\neq \emptyset$ one has $d_{1}=d_{2}$.

If a group $\Gamma$ acts discontinuously on a topological space $E$, then the quotient map $q:E\rightarrow \Gamma \backslash E$ with $q(e)=\Gamma e$ is a normal covering. $^{p.72}$ Hereby $\Gamma \backslash E=\{\Gamma e:e\in E\}$ is the quotient space and $\Gamma e=\{\gamma (e):\gamma \in \Gamma \}$ is the orbit of the group action.

Examples

• The covering $q\colon S^{1}\to S^{1}$ with $q(z)=z^{n}$ is a normal coverings for every $n\in \mathbb {N}$.
• Every simply connected covering is a normal covering.

Calculation

Let $\Gamma$ be a group, which acts discontinuously on a topological space $E$ and let $q:E\rightarrow \Gamma \backslash E$ be the normal covering.

• If $E$ is path-connected, then $Deck(q)\cong \Gamma$. $^{p.72}$

• If $E$ is simply connected, then $Deck(q)\cong \pi _{1}(X)$. $^{p.71}$

Examples

• Let $n\in \mathbb {N}$. The antipodal map $g:S^{n}\rightarrow S^{n}$ with $g(x)=-x$ generates, together with the composition of maps, a group $D(g)\cong \mathbb {Z/2Z}$ and induces a group action $D(g)\times S^{n}\rightarrow S^{n},(g,x)\mapsto g(x)$, which acts discontinuously on $S^{n}$. Because of $\mathbb {Z_{2}} \backslash S^{n}\cong \mathbb {R} P^{n}$ it follows, that the quotient map $q\colon S^{n}\rightarrow \mathbb {Z_{2}} \backslash S^{n}\cong \mathbb {R} P^{n}$ is a normal covering and for $n>1$ a universal covering, hence $Deck(q)\cong \mathbb {Z/2Z} \cong \pi _{1}({\mathbb {R} P^{n}})$ for $n>1$.
• Let $SO(3)$ be the special orthogonal group, then the map $f:SU(2)\rightarrow SO(3)\cong \mathbb {Z_{2}} \backslash SU(2)$ is a normal covering and because of $SU(2)\cong S^{3}$, it is the universal covering, hence $Deck(f)\cong \mathbb {Z/2Z} \cong \pi _{1}(SO(3))$.
• With the group action $(z_{1},z_{2})*(x,y)=(z_{1}+(-1)^{z_{2}}x,z_{2}+y)$ of $\mathbb {Z^{2}}$ on $\mathbb {R^{2}}$, whereby $(\mathbb {Z^{2}} ,*)$ is the semidirect product $\mathbb {Z} \rtimes \mathbb {Z}$, one gets the universal covering $f:\mathbb {R^{2}} \rightarrow (\mathbb {Z} \rtimes \mathbb {Z} )\backslash \mathbb {R^{2}} \cong K$ of the klein bottle $K$, hence $Deck(f)\cong \mathbb {Z} \rtimes \mathbb {Z} \cong \pi _{1}(K)$.
• Let $T=S^{1}\times S^{1}$ be the torus which is embedded in the $\mathbb {C^{2}}$. Then one gets a homeomorphism $\alpha :T\rightarrow T:(e^{ix},e^{iy})\mapsto (e^{i(x+\pi )},e^{-iy})$, which induces a discontinuous group action $G_{\alpha }\times T\rightarrow T$, whereby $G_{\alpha }\cong \mathbb {Z/2Z}$. It follows, that the map $f:T\rightarrow G_{\alpha }\backslash T\cong K$ is a normal covering of the klein bottle, hence $Deck(f)\cong \mathbb {Z/2Z}$.
• Let $S^{3}$ be embedded in the $\mathbb {C^{2}}$. Since the group action $S^{3}\times \mathbb {Z/pZ} \rightarrow S^{3}:((z_{1},z_{2}),[k])\mapsto (e^{2\pi ik/p}z_{1},e^{2\pi ikq/p}z_{2})$ is discontinuously, whereby $p,q\in \mathbb {N}$ are coprime, the map $f:S^{3}\rightarrow \mathbb {Z_{p}} \backslash S^{3}=:L_{p,q}$ is the universal covering of the lens space $L_{p,q}$, hence $Deck(f)\cong \mathbb {Z/pZ} \cong \pi _{1}(L_{p,q})$.

Galois correspondence

Let $X$ be a connected and locally simply connected space, then for every subgroup $H\subseteq \pi _{1}(X)$ there exists a path-connected covering $\alpha :X_{H}\rightarrow X$ with $\alpha _{\#}(\pi _{1}(X_{H}))=H$. $^{p.66}$

Let $p_{1}:E\rightarrow X$ and $p_{2}:E'\rightarrow X$ be two path-connected coverings, then they are equivalent iff the subgroups $H=p_{1\#}(\pi _{1}(E))$ and $H'=p_{2\#}(\pi _{1}(E'))$ are conjugate to each other. $^{p.482}$

Let $X$ be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:

${\begin{matrix}\qquad \displaystyle \{{\text{Subgroup of }}\pi _{1}(X)\}&\longleftrightarrow &\displaystyle \{{\text{path-connected covering }}p:E\rightarrow X\}\\H&\longrightarrow &\alpha :X_{H}\rightarrow X\\p_{\#}(\pi _{1}(E))&\longleftarrow &p\\\displaystyle \{{\text{normal subgroup of }}\pi _{1}(X)\}&\longleftrightarrow &\displaystyle \{{\text{normal covering }}p:E\rightarrow X\}\\H&\longrightarrow &\alpha :X_{H}\rightarrow X\\p_{\#}(\pi _{1}(E))&\longleftarrow &p\end{matrix}}$

For a sequence of subgroups $\displaystyle \{{\text{e}}\}\subset H\subset G\subset \pi _{1}(X)$ one gets a sequence of coverings ${\tilde {X}}\longrightarrow X_{H}\cong H\backslash {\tilde {X}}\longrightarrow X_{G}\cong G\backslash {\tilde {X}}\longrightarrow X\cong \pi _{1}(X)\backslash {\tilde {X}}$. For a subgroup $H\subset \pi _{1}(X)$ with index $\displaystyle [\pi _{1}(X):H]=d$, the covering $\alpha :X_{H}\rightarrow X$ has degree $d$.

Classification

Definitions

Category of coverings

Let $X$ be a topological space. The objects of the category ${\boldsymbol {Cov(X)}}$ are the coverings $p:E\rightarrow X$ of $X$ and the morphisms between two coverings $p:E\rightarrow X$ and $q:F\rightarrow X$ are continuous maps $f:E\rightarrow F$, s.t. the diagram

commutes.

G-Set

Let $G$ be a topological group. The category ${\boldsymbol {G-Set}}$ is the category of sets which are G-sets. The morphisms are G-maps $\phi :X\rightarrow Y$ between G-sets. They satisfy the condition $\phi (gx)=g\,\phi (x)$ for every $g\in G$.

Equivalence

Let $X$ be a connected and locally simply connected space, $x\in X$ and $G=\pi _{1}(X,x)$ be the fundamental group of $X$. Since $G$ defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor $F:{\boldsymbol {Cov(X)}}\longrightarrow {\boldsymbol {G-Set}}:p\mapsto p^{-1}(x)$ is an equivalence of categories. $^{p.68-70}$

Literatur

• Allen Hatcher: Algebraic Topology. Cambridge Univ. Press, Cambridge, ISBN 0-521-79160-X
• Otto Forster: Lectures on Riemann surfaces. Springer Berlin, München 1991, ISBN 978-3-540-90617-9
• James Munkres: Topology. Upper Saddle River, NJ: Prentice Hall, Inc., ©2000, ISBN 978-0-13-468951-7
• Wolfgang Kühnel: Matrizen und Lie-Gruppen. Springer Fachmedien Wiesbaden GmbH, Stuttgart, ISBN 978-3-8348-9905-7
• Maximiliano Aguilar and Miguel Socolovsky: The Universal Covering Group of U(n) and Projective Representations. Hrsg.: International Journal of Theoretical Physics. Dezember 1999