Covers, subgroups $\Gamma \subseteq {\pi }_1\left(U\right)$ and ramification

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

Last update: 14 June 2012

Covers and subgroups $\Gamma \subseteq {\pi }_1\left(U\right)$

Let $U$ be a topological space.

A path from $x$ to $y$ in $U$ is a continuous map $$\gamma :[0,1]\to Usuch\; that\gamma \left(0\right)=xand\gamma \left(1\right)=y.$$ The space $U$ is path connected if there is a path ${\gamma }_{xy}:[0,1]\to U$ with $\gamma \left(0\right)=x$ and $\gamma \left(1\right)=y,$ for all $x,y\in U.$

The space $U$ is contractible if the identity map $$\begin{array}{rcl}U& \to & U\\ u& \mapsto & u\end{array}$$ is homotopic to a constant map.

Let $U$ be a path connected topological space such that every point has a contractible neighbourhood. A covering of $U$ is a pair $(V,p:V\to U)$ where $V$ is a path connected topological space such that every point has a contractible neighbourhood and $$p:V\to Uis\; a\; continuous\; map$$ such that

1. For every $u\in U,$ $p^{-1}\left(u\right)$ is discrete
2. If $u\in U$ then there is an open neighbourhood $N$ of $u$ and a homeomorphism $$\phi :p^{-1}\left(N\right)\to N\times p^{-1}\left(u\right)$$ such that $p={\pi }_1\circ \phi$ where $$\begin{array}{rrcl}{\pi }_1:& N\times p^{-1}\left(u\right)& \to & N\\ & (n,x)& \mapsto & n.\end{array}$$

Let $u_0\in U,$ i.e. fix a "base point" $u_0\in U.$

The universal covering space $\tilde{U}$ of $U$ is $$\tilde{U}=\frac{\{k:[0,1]\to U|\genfrac{}{}{0ex}{}{his\; continuous}{h\left(0\right)=u_0}\phantom{|}\}}{⟨homotopy{h}_{u_{0}x}\sim h_{u_{0}x}^{\prime }⟩}$$ with projection given by $$\begin{array}{rrcl}p:& \tilde{U}& \to & U\\ & h& \mapsto & h\left(1\right).\end{array}$$

The fundamental group of $U$ is $${\pi }_1\left(U\right)=\frac{\{h:[0,1]\to U|\genfrac{}{}{0ex}{}{his\; continuous}{h\left(0\right)=h\left(1\right)=u_0}\phantom{|}\}}{⟨homotopy{h}_{u_0,u_0}^{\prime }\sim h_{u_0,u_0}⟩}$$ and multiplication given by $${\gamma }_1{\gamma }_2\left(t\right)=\{\begin{array}{ll}{\gamma }_2\left(2t\right),& if0\le t\le \frac{1}{2},\\ {\gamma }_1(2t-1),& if\frac{1}{2}\le t\le 1,\end{array}\phantom{\}}$$ for ${\gamma }_1,{\gamma }_2\in {\pi }_1\left(U\right).$

The group ${\pi }_1\left(U\right)$ acts on $\tilde{U}$ by $$\gamma h\left(t\right)=\{\begin{array}{ll}\gamma \left(2t\right),& if0\le t\le \frac{1}{2},\\ h(2t-1),& if\frac{1}{2}\le t\le 1,\end{array}\phantom{\}}$$ for $\gamma \in {\pi }_1\left(U\right), h\in \tilde{U}.$

A covering $p:V\to U$ is finite if $p^{-1}\left(u\right)$ is finite for all $u\in U.$

A covering $p:V\to U$ is regular if ${\pi }_1\left(V\right)$ is a normal subgroup of ${\pi }_1\left(U\right).$

Note that ${\pi }_1\left(V\right)\stackrel{p}{\&hkrarr;}{\pi }_1\left(U\right)$ since if $$h:[0,1]\to Vthenp\circ h:[0,1]\stackrel{h}{\to }V\stackrel{p}{\to }U.$$

Coverings $p_1:V_1\to U$ and $p_2:V_2\to U$ are equivalent if there is a homeomorphism $\phi :V_1\to V_2$ such that $p_2\circ \phi =p_1,$

Subgroups ${\Gamma }_1$ and ${\Gamma }_2$ of ${\pi }_1\left(U\right)$ are conjugate if there is a $\gamma \in {\pi }_1\left(U\right)$ such that $\gamma {\Gamma }_1{\gamma }^{-1}={\Gamma }_2.$

There is a bijection $$\begin{array}{rcl}\frac{\left\{\genfrac{}{}{0ex}{}{coverings}{p:V\to U}\right\}}{⟨equivalence⟩}& \stackrel{1-1}{\leftrightarrow }& \frac{\left\{\genfrac{}{}{0ex}{}{subgroups\Gamma }{of{\pi }_1\left(U\right)}\right\}}{⟨conjugacy⟩}\\ p:V\to U& \mapsto & {\pi }_1\left(V\right)\\ \begin{array}{rcl}\Gamma \backslash \tilde{U}& \to & U\\ \Gamma k& \mapsto & k\left(1\right)\end{array}& ↤& \Gamma \end{array}$$

Ramification

A Riemann surface is a 1-dimensional complex manifold.

Let $X$ be a connected compact Riemann surface. Then there is a homeomorphism $$X\stackrel{\sim }{\to }\begin{array}{c} \end{array}$$ where the number of holes is the genus of $X,$ $$g=\frac{1}{2}rk(H\prime \left(X\right)).$$ Let $X$ and $Y$ be compact Riemann surfaces and let $$f:X\to Y \; be\; a\; holomorphic\; map.$$ The function field of $X$ is The map $f:X\to Y$ induces a field homomorphism $$\begin{array}{rrcl}{f}^{*}:& ℳ\left(Y\right)& \to & ℳ\left(X\right)\\ & \phi & \mapsto & \phi \circ f\end{array}$$ The degree of $f:X\to Y$ is $\mathrm{deg}\left(f\right)=[ℳ\left(X\right):ℳ\left(Y\right)].$ A generic point of $Y$ is a $y\in Y$ such that $$\mathrm{Card}\left(f^{-1}\left(y\right)\right)=\mathrm{deg}\left(f\right).$$ A ramification point, or branch point, is a $y\in Y$ such that $$\mathrm{Card}\left(f^{-1}\left(y\right)\right)<\mathrm{deg}\left(f\right).$$ Let $y\in Y$ and let $f^{-1}\left(y\right)=\{x_1,...,x_r\}.$ Let $\eta \in Y$ be a generic point near $y.$ The ramification index of $x_i$ is $$e_i=\left(number\; of\; elements\; of{f}^{-1}\left(\eta \right)near{x}_i\right)$$ The point $x_i$ is ramified if $e_i>1.$ We have $$\sum _{i=1}^r{e}_i=\mathrm{deg}\left(f\right).$$

Let $Y$ be a compact Riemann surface. There is bijection $$\begin{array}{rcl}\left\{\genfrac{}{}{0ex}{}{finite\; coverings}{p:V\to Y-\left(finite\; no.\; of\; pts\right)}\right\}& \stackrel{1-1}{\leftrightarrow }& \left\{\genfrac{}{}{0ex}{}{holomorphic\; mapsf:X\to Y,where}{Xis\; a\; compact\; Riemann\; surface}\right\}\\ p:V\to Y-\{y_1,...,y_r\}& \mapsto & \hat{p}:\hat{V}\to Y\\ f:f^{-1}(Y-\left\{\genfrac{}{}{0ex}{}{ramification}{points\; off}\right\})\to Y-\left\{\genfrac{}{}{0ex}{}{ram.\; pts}{off}\right\}& ↤& f:X\to Y\end{array}$$ where $\hat{V}$ is a compactification of $V.$

The composite map $$\left\{\genfrac{}{}{0ex}{}{finite\; covers}{p:V\to \{y_1,...,y_r\}}\right\}\to \left\{\genfrac{}{}{0ex}{}{holomorphic}{f:X\to Y}\right\}\to \left\{\genfrac{}{}{0ex}{}{field\; extensions}{f^{*}:ℳ\left(Y\right)\to ℳ\left(X\right)}\right\}$$ sends regular covers to Galois extensions.

Notes and References

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References

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