## Covers, subgroups $\Gamma \subseteq {\pi }_{1}\left(U\right)$ and ramification

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 $γ: [0,1]→U such that γ(0) = x and γ(1) =y.$ The space $U$ is path connected if there is a path ${\gamma }_{xy}:\left[0,1\right]\to U$ 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 $U→U u↦u$ 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 $\left(V,p:V\to U\right)$ where $V$ is a path connected topological space such that every point has a contractible neighbourhood and $p:V→U is a continuous map$ 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 $φ: p-1(N) → N×p-1(u)$ such that $p={\pi }_{1}\circ \phi$ where $π1: N×p-1(u) → N (n,x) ↦ n.$

Let ${u}_{0}\in U,$ i.e. fix a "base point" ${u}_{0}\in U.$

The universal covering space $\stackrel{˜}{U}$ of $U$ is $U˜ = { k:[0,1]→U | his continuous h(0)=u0 | } ⟨ homotopy hu0x ∼ hu0x′ ⟩$ with projection given by $p: U˜ → U h ↦ h(1).$

The fundamental group of $U$ is $π1(U) = { h:[0,1]→U | h is continuous h(0)=h(1)=u0 | } ⟨ homotopy hu0,u0′ ∼ hu0,u0 ⟩$ and multiplication given by $γ1γ2(t) = { γ2(2t), if 0≤t≤12, γ1(2t-1), if 12≤t≤1, }$ for ${\gamma }_{1},{\gamma }_{2}\in {\pi }_{1}\left(U\right).$

The group ${\pi }_{1}\left(U\right)$ acts on $\stackrel{˜}{U}$ by $γh(t) = { γ(2t), if 0≤t≤12, h(2t-1), if 12≤t≤1, }$ for

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 $h:[0,1]→V then p∘h: [0,1] →h V →p U.$

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 ${ coverings p:V→U } ⟨ equivalence ⟩ ↔1-1 { subgroups Γ of π1(U) } ⟨ conjugacy ⟩ p:V→U ↦ π1(V) Γ\U˜ → U Γk ↦ k(1) ↤ Γ$

## Ramification

A Riemann surface is a 1-dimensional complex manifold.

Let $X$ be a connected compact Riemann surface. Then there is a homeomorphism $X →∼$ where the number of holes is the genus of $X,$ $g = 12rk ( H′(X) ).$ Let $X$ and $Y$ be compact Riemann surfaces and let The function field of $X$ is $ℳ(X) = { meromorphic functions φ:X→ℂ }.$ The map $f:X\to Y$ induces a field homomorphism $f*: ℳ(Y) → ℳ(X) φ ↦ φ∘f$ The degree of $f:X\to Y$ is $\mathrm{deg}\left(f\right)=\left[ℳ\left(X\right):ℳ\left(Y\right)\right].$ A generic point of $Y$ is a $y\in Y$ such that $Card(f-1(y)) = deg(f).$ A ramification point, or branch point, is a $y\in Y$ such that $Card(f-1(y)) < deg(f).$ Let $y\in Y$ and let ${f}^{-1}\left(y\right)=\left\{{x}_{1},...,{x}_{r}\right\}.$ Let $\eta \in Y$ be a generic point near $y.$ The ramification index of ${x}_{i}$ is $ei = ( number of elements of f-1(η) near xi )$ The point ${x}_{i}$ is ramified if ${e}_{i}>1.$ We have $∑i=1r ei = deg(f).$

Let $Y$ be a compact Riemann surface. There is bijection ${ finite coverings p:V→Y- ( finite no. of pts ) } ↔1-1 { holomorphic maps f:X→Y, where X is a compact Riemann surface } p:V→Y-{ y1,...,yr } ↦ p^: V^→Y f: f-1( Y-{ ramification points of f } ) → Y-{ ram. pts of f } ↤ f:X→Y$ where $\stackrel{^}{V}$ is a compactification of $V.$

The composite map ${ finite covers p:V→ {y1,...,yr} } → { holomorphic f:X→Y } → { field extensions f*: ℳ(Y) → ℳ(X) }$ sends regular covers to Galois extensions.

## Notes and References

Where are these from?

References?