Foundations of the Theory of Klein Surfaces
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219 Norman L. Alling
The University of Rochester, Rochester, NYIUSA
Newcomb Greenleaf
The University of Texas, Austin, Tx/USA
Foundations of the Theory of Klein Surfaces
Springer-Verlag Berlin· Heidelberg· NewYork 1971
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zurich
219 Norman L. Alling
The University of Rochester, Rochester, NYIUSA
Newcomb Greenleaf
The University of Texas, Austin, Tx/USA
Foundations of the Theory of Klein Surfaces
Springer-Verlag Berlin· Heidelberg· NewYork 1971
AMS Subject Classifications (1970): 14H05, 14)25, 30A46
ISBN 3-540-05577-0 Springer-Verlag Berlin· Heidelberg· New York ISBN 0-387-05577-0 Springer-Verlag New York· Heidelberg' Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 73-172693. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach
INTRODUCTION
It has long been known that the category
of compact Rie-
mann surfaces and non-constant analytic maps, and the category of complex-algebraic function fields and complex isomorphisms are, via two contravariant functions, coequivalent; thus an analytic theory and an algebraic theory are tied together. While investigating several Banach algebras on compact Riemann surfaces
I
with non-empty boundary
oX
([A Z]' [A [A the 4]), 3],
first author posed the following question for himself:
what is the
simplest algebraic object which can be associated with
I,
which let that
can be recovered?
I
E(I)
The answer seems to be the following:
be the field of all functions
f(oX)
R U [ro}.
surface
I
f
I
meromorphic on
such
This field is an algebraic function field
in one variable over the reals. verse question:
from
It is natural then to ask the con-
given such a field
E,
is there a compact Riemann
(possibly with boundary) such that
answer to this question, interestingly, is no.
E
E(1)?
The
The following field,
long known to algebraic geometers, supplies a counter-example to such a conjecture:
1 et
E = R( x,y ),
where
X
z+
y2
=
-1.
The
present collaboration began at this juncture. Let
be the category of all real-algebraic function fields
and all real-linear isomorphisms.
Given such a field E,
braic geometers have long known how to associate a curve E;
for example let
X;: {o:
0
a valuation ring of
E
the algeX with over
R}.
The usual topology put on such a curve is the Zariski topology in
IV which the proper closed sets are the finite sets. does not utilize the topology on fold.
Such a to
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