Homology of Locally Semialgebraic Spaces
Locally semialgebraic spaces serve as an appropriate framework for studying the topological properties of varieties and semialgebraic sets over a real closed field. This book contributes to the fundamental theory of semialgebraic topology and falls into t
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1484
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
1484
Hans Delfs
Homology of Locally Semialgebraic Spaces
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Hans Delfs Fakultat fur Mathematik Universitat Regensburg Universitatsstrabe 31 W-8400 Regensburg, FRG
Mathematics Subject Classification (1991): 14G30, 14F45, 55N30, 55N35, 14C 17, 54DI8
ISBN 3-540-54615-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54615-4 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned. specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper
Introduction
The basic task in real algebraic (or better semialgebraic) geometry is to study the set of solutions of a finite system of polynomial inequalities over a real closed field R. Such a set is called a semialgebraic set over R. The semialgebraic subsets of H" are obtained from the basic open sets
U(P)
=
{x ERn I P(x) > O},
where P E R[X1 , . . . ,Xnl is a polynomial, by finitely many applications of the set theoretic operations of uniting, intersecting and complementing. The interval topology of R induces a topology on every semialgebraic set over R called strong topology. Unfortunately the arising topological spaces are totally disconnected, except in the case R = R. This pathology can be remedied. Strong topology is replaced by semialgebraic topology, a topology in the sense of Grothendieck (d. [AJ): Only open semialgebraic subsets are admitted as "open sets", and essentially only coverings by finitely many open semialgebraic subsets are admitted as "open coverings". These restricted topological spaces are the basic objects studied in semialgebraic topology. (For details concerning this and other notions from semialgebraic geometry we refer to [Br], [BCR], [DK], [DK3], [DK4J). It is easier to study not only semialgebraic sets which are embedded in some algebraic variety over R but to study more generally spaces which, locally, look like a semialgebraic set. This observation leads to the notion of semialgebraic spaces. An affine semialgebraic space over R is a ringed space which is isomorphic to a semialgebraic subset N of some affine R-variety V equipped with its sheaf ON of semialgebraic functions (d. [DKJ). N is considered in its semialgebraic top
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