Covers, subgroups Γπ1(U) and ramification

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 14 June 2012

Covers and subgroups Γπ1(U)

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 γxy: [0,1]U with γ(0) = x and γ(1) = y, for all x,yU.

The space U is contractible if the identity map UU uu 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:VU) where V is a path connected topological space such that every point has a contractible neighbourhood and p:VU is a continuous map such that

  1. For every uU, p-1(u) is discrete
  2. If uU then there is an open neighbourhood N of u and a homeomorphism φ: p-1(N) N×p-1(u) such that p = π1φ where π1: N×p-1(u) N (n,x) n.

Let u0U, i.e. fix a "base point" u0U.

The universal covering space 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 0t12, γ1(2t-1), if 12t1, } for γ1, γ2 π1(U).

The group π1(U) acts on U˜ by γh(t) = { γ(2t), if 0t12, h(2t-1), if 12t1, } for γπ1(U),   hU˜.

A covering p:VU is finite if p-1(u) is finite for all uU.

A covering p:VU is regular if π1(V) is a normal subgroup of π1(U).

Note that π1(V) &hkrarr;p π1(U) since if h:[0,1]V then ph: [0,1] h V p U.

Coverings p1: V1U and p2: V2U are equivalent if there is a homeomorphism φ: V1V2 such that p2φ = p1,

V1 V2 U φ p1 p2

Subgroups Γ1 and Γ2 of π1(U) are conjugate if there is a γπ1(U) such that γΓ1γ-1 = Γ2.

There is a bijection { coverings p:VU } equivalence 1-1 { subgroups Γ of π1(U) } conjugacy p:VU π1(V) Γ\U˜ U Γk k(1) Γ


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 f:XY   be a holomorphic map. The function field of X is (X) = { meromorphic functions φ:X }. The map f:XY induces a field homomorphism f*: (Y) (X) φ φf The degree of f:XY is deg(f) = [(X):(Y)]. A generic point of Y is a yY such that Card(f-1(y)) = deg(f). A ramification point, or branch point, is a yY such that Card(f-1(y)) < deg(f). Let yY and let f-1(y) = { x1,...,xr }. Let ηY be a generic point near y. The ramification index of xi is ei = ( number of elements of f-1(η) near xi ) The point xi is ramified if ei>1. We have i=1r ei = deg(f).

ramified here y ramified point X f Y degf=2 x1 x2 X Y deg(f)=4 = #   of sheets 2 sheets come together here e1=2 2 sheets come together here e2=2

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

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

Notes and References

Where are these from?



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