Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
Last update: 14 June 2012
Covers and subgroups
Let be a topological space.
A path from to in is a continuous map
The space is path connected if there is a path
The space is contractible if the identity map
is homotopic to a constant map.
Let be a path connected topological space such that every point has a contractible neighbourhood. A covering of is a pair
where is a path connected topological space such that every point has a contractible neighbourhood and
For every is discrete
If then there is an open neighbourhood of and a homeomorphism
Let i.e. fix a "base point"
The universal covering space of is
with projection given by
The fundamental group of is
and multiplication given by
The group acts on by
A covering is finite if is finite for all
A covering is regular if is a normal subgroup of
are equivalent if there is a homeomorphism
Subgroups and of are conjugate if there is a
There is a bijection
A Riemann surface is a 1-dimensional complex manifold.
Let be a connected compact Riemann surface. Then there is a homeomorphism
where the number of holes is the genus of
Let and be compact Riemann surfaces and let
The function field of is
The map induces a field homomorphism
The degree of is
A generic point of is a such that
A ramification point, or branch point, is a such that
Let and let
Let be a generic point near The ramification index of is
The point is ramified if We have
Let be a compact Riemann surface. There is bijection
where is a compactification of
The composite map
sends regular covers to Galois extensions.