Topology of Singular Fibers of Differentiable Maps
The volume develops a thorough theory of singular fibers of generic differentiable maps. This is the first work that establishes the foundational framework of the global study of singular differentiable maps of negative codimension from the viewpoint of d
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Osamu Saeki
Topology of Singular Fibers of Differentiable Maps
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Author Osamu Saeki Faculty of Mathematics Kyushu University Hakozaki, Fukuoka 812-8581, Japan e-mail: [email protected]
Library of Congress Control Number: 2004110903 Mathematics Subject Classification (2000): 57R45, 57N13
ISSN 0075-8434 ISBN 3-540-23021-1 Springer Berlin Heidelberg New York DOI 10.1007/b100393 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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. Violations are liable to prosecution under the German Copyright Law. Springer is part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author 41/3142-543210 - Printed on acid-free paper
To C´elia
Preface
In 1999, a friend of mine, Kazuhiro Sakuma, kindly asked me to give a series of lectures in the Kwansai Seminar on Differential Analysis, held at the Kinki University, Japan. At that time, I was studying the global topology of differentiable maps of 4-dimensional manifolds into lower dimensional manifolds. Sakuma and I had obtained a lot of interesting results concerning the relationship between the singularities of such maps and the differentiable structures of 4-dimensional manifolds; however, our results were not based on a systematic theory and were not satisfactory in a certain sense. So I was trying to construct such a systematic theory when I was asked to give lectures. I wondered what kind of objects can reflect the global properties of manifolds. “Singularity” of a differentiable map can be such an object, but it is local in nature. I already knew that the notion of the Stein factorization played an important role in the global study of such maps; for example, refer to the works of Burlet–de Rham [7] or Kushner–Levine–Porto [28, 30]. Stein factorization is constructed by considering the connected components of the fibers of a given map. This inspired me to consider singular fibers of differentiable maps. I promptly started the classification of singular fibers of stable maps of orientable 4-manifolds into 3-manifolds. It was not a difficult task, though quite tedious. Then I obtained the modulo two Euler characteristic formula in terms of the number of a certain singular fiber, by using Sz˝ ucs’s formula [55], which Nu˜ no Ballesteros and I
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