From: [Permission pending]
Subject: Re: Fundameltal group of Riemann surfaces
To: rusin@math.niu.edu (Dave Rusin)
Date: Fri, 18 Nov 1994 08:16:30 +0000 (MET)
Hi Dave,
> If you mean compact connected Riemann surfaces, these are the same as
> n-holed orientable surfaces; the fundamental group is generated by
> 2n generators xi and yi (i=1,...,n) subject to the single relation
> that the product of the commutators is 1 (x1^-1 y1^-1 x1 y1 x2^-1 etc.)
Thank you, that's exactly what I need :-)
> (Actually the Seifert-VanKampen theorem proves
> the structure of the fundamental group is exactly what I said).
Could you tell me any book, where I can find this theorem?
[sig deleted -- djr]
==============================================================================
Date: Fri, 18 Nov 94 11:37:55 CST
From: rusin (Dave Rusin)
To: [Permission pending]
Subject: Re: Fundameltal group of Riemann surfaces
The VanKampen theorem ought to be in most books on algebraic topology.
I'm pretty sure, for example, that it's in Massey's "Algebraic Topology,
an Introduction". The main difficulty in the theorem is really in
setting up the algebraic preliminaries to state it; most texts give
one or more simplified versions which are easier to state (if not
prove). For example, one version is:
If X = U u V and U intersect V is simply connected and connected, then
pi_1(X) is the free product of pi_1(U) and pi_1(V).
The version you'll need to prove the result I sent you before says that
more generally (when W := U intersect V is not simply connected) is
that pi_1(X) is the above free product modulo the normal subgroup
generated by the common images of elements of pi_1(W). (See what I
mean about the difficulty of preparing the reader with sufficient
algebra?) In practice, you write down the union of the generators of
pi_1(U) and pi_1(V), then for relations you take the relations of
each, together with a collection of relations a(g)=b(g), one
for each generator g of pi_1(W), where a and b are the
homomorphisms on pi_1 induced by the inclusions of W into
U and V respectively.
Taking U to be the interior of the polygon I mentioned and V to be
a neighborhood of the edge [oops -- I forgot to say U and V have to
be open in all the above], we see that U is contractible, W is
an open annulus, and V, after performing the necessary identifications,
has the loops as a deformation retract (so that its fundamental group
is that of a bouquet of n circles; by induction the Seifert-VanKampen
theorem can be applied here too to show pi_1(V) is the free group on
n generators).
let me know if this is still unclear
dave