Geometry of Subanalytic and Semialgebraic Sets
Real analytic sets in Euclidean space (Le. , sets defined locally at each point of Euclidean space by the vanishing of an analytic function) were first investigated in the 1950's by H. Cartan [Car], H. Whitney [WI-3], F. Bruhat [W-B] and others. Their app
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Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
Masahiro Shiota
Geometry of Subanalytic and Semialgebraic Sets
Springer Science+Business Media, LLC
Masahiro Shiota Department of Mathematics Nagoya University Furocho, Chikusa Nagoya 464, Japan
Library of Congress Cataloging-In-Publication Data Shiota, Masahiro, 1947Geometry of subanalytic and semialgebraic sets / Masahiro Shiota. p. cm. -- (Progress in mathematics; v. 150) Includes bibliographical references (p. - ) and index. ISBN 978-1-4612-2008-4 (eBook) ISBN 978-1-4612-7378-3 DOI 10.1007/978-1-4612-2008-4 1. Semialgebraic sets. 2. Semianalytic sets. 1. Title. II. Series: Progress in mathematics (Boston, Mass.) ; voI. 150. QA564.S46 1997 97-9061 516.3--dc21 CIP
AMS Classitication: 03c, 14p, 32c, 32s,58a Printed on acid-free paper © 1997 Springer Science+Business Media New York Origina11y published by Birkhauser Boston in 1997 Softcover reprint of the hardcover 1st edition 1997
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CONTENTS
Introduction ..................................................... vii CHAPTER I.
PRELIMINARIES .................................. 1
§I.L Whitney stratifications ..................................... 1 §I.2. Subanalytic sets and semialgebraic sets ...................... 40 §I.3. PL topology and Coo triangulations ......................... 54 CHAPTER II.
X-SETS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
§ILL X-sets ................................................... 97 §IL2. Triangulations of X-sets .................................. 116 §II.3. Triangulations of X-functions .............................. 131 §II.4. Triangulations of semialgebraic and Xo sets and functions ..... 146 §II.5. C r X-manifolds .......................................... 156 §II.6. X-triviality of X-maps .................................... 186 §II.7. X-singularity theory ...................................... 235 CHAPTER III.
HAUPTVERMUTUNG FOR POLyHEDRA ....... 270
§IILL Certain conditions for two polyhedra to be PL homeomorphic. 270 §IIL2. Proofs of Theorems IILL1 and III.L2 ...................... 274 CHAPTER IV.
TRIANGULATIONS OF X-MAPS ............... 305
§IV.L Conditions fo
