Stratified Morse Theory
Due to the lack of proper bibliographical sources stratification theory seems to be a "mysterious" subject in contemporary mathematics. This book contains a complete and elementary survey - including an extended bibliography - on stratification theory, in
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Editorial Board E. Bombieri, Princeton S. Feferman, Stanford N.H. Kuiper, Bures-sur-Yvette P.Lax, NewYork H.W.Lensira,Jr., Berkeley R. Remmert (Managing Editor), Munster W. Schmid, Cambridge, Mass. J-P. Serre, Paris J. Tits, Paris K.K. Uhlenbeck, Austin
Mark Goresky Robert MacPherson
Stratified Morse Theory
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Mark Goresky Department of Mathematics Northeastern University Boston, MA 02115, USA Robert MacPherson Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139, USA
Mathematics Subject Classification (1980): 14F35, 57D70
ISBN-13: 978-3-642-71716-1 e-ISBN-13: 978-3-642-71714-7 DOT: 10.1007/978-3-642-71714-7
Library of Congress Cataloging in Publication Data Goresky, Mark, 1950--. Stratified Morse theory. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 14) Bibliography: p. Includes index. 1. Morse theory. 2. Analytic spaces. 3. Topology. I. MacPherson, Robert, 1944-. II. Title. III. Series. QA331.G655 1988 515.7'3 87-26418
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To Rene Thorn who contributed three essential steps to the story presented here - the idea that a Morse function leads to a cell decomposition (1949) - the idea of studying complex varieties using Morse theory (1957) - the isotopy lemmas of stratification theory (1969)
Preface
This book explores the natural generalization of Morse theory to stratified spaces. Applications are given, primarily to the topology of complex analytic varieties. The main theorems are proven here for the first time, although they are heirs to a long line of historical development (see Sect. 1.7 and Sect. 2.8 of the introduction and Sect. 1.0 and Sect. 2.0 of Part I). The work presented here was first announced in 1980 [GM1]. The original proofs were discouragingly complicated and technical. During the intervening years we have developed methods which have greatly simplified the arguments, while at the same time making them more geometric (see Chap. 4 and Chap. 5 of Part I). Conversations with P. Deligne, R. Lazarsfeld, and especially W. Fulton about potential applications to complex varieties were instrumental in persuading us to take up the study of stratified Morse theory. We have also profited from valuable conversations with G. Bartel, L. Kaup, P. Orlik, P. Schapira, B. Teissier, R. Thorn, R. Thomason, K. Vilonen, H. Hamm, Le D.T., D. Massey,
