From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Topology of C
Date: 31 Mar 1997 22:25:32 GMT

In article <5hdq8u$lau$4@rzsun02.rrz.uni-hamburg.de>,
Hauke Reddmann <fc3a501@AMRISC04.math.uni-hamburg.de> wrote:
>C (with oo) has the topology of a sphere. I guess
>a lot of statements about C depend on this fact,
>like the application of Brouwers fix-point theorem.
>
>Can you tell me some?

Arguably most global theorems use this at least indirectly. See e.g. the
latter chapters of Conway's  "Functions of One Complex Variable". Also
of interest are what are essentially applications of degree theory in
differential topology, such as e.g. the Fundamental Theorem of Algebra.
Nice reference: Milnor's "Topology from the Differentiable Viewpoint".

>Are there well-known other sets of numbers with,
>say, a torus topology (angles modulo 2pi or the
>like) AND some decent algebraic structure?

Well, there's plenty of algebra involved in Riemann surface theory
(e.g. the period matrix and so on). But if you want a topological space
which is also a group (such that the group operations are continuous
with respect to the topology) then the fundamental group would have to
be abelian. Among the manifolds of dimension at most 2, this alone rules
out much algebraic structure on most of them. Indeed, the only topological
groups which are also connected manifolds of dimension at most 2 are
the point, the line, the circle, the plane, the once-punctured plane, and
the torus. All of them are probably already familiar to you, although
the tori are perhaps the most interesting, as there are many group
structures possible on the torus which are inequivalent as complex
manifolds (that is, there are non-isomorphic elliptic curves).

dave