Notation, Conventions and Other Preliminaries
The main goal of this chapter is to introduce some notation and terminology. We assume that the reader is more or less familiar with the basic concepts of algebraic topology (homotopy and homology). Typical references are: tom Dieck–Kamps–Puppe [1], tom D
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Yuli B. Rudyak
On Thom Spectra, Orientability, and Cobordism
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Yuli B. Rudyak Department of Mathematics University of Florida 358 Little Hall PO Box 118105 Gainesville, FL 32611–8105 USA e-mail: [email protected]fl.edu
Corrected 2nd printing 2008 ISBN 978-3-540-62043-3 Springer Mongraphs in Mathematics ISSN 1439-7382 Library of Congress Control Number: 97032730 Mathematics Subject Classification (1991): 55Nxx, 55Rxx, 55Sxx, 57Nxx, 57Qxx, 57Rxx © 1998 Springer-Verlag 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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 for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com
Dedicated to my parents
Foreword
For many years, Algebraic Topology rests on three legs: “ordinary” Cohomology, K-theory, and Cobordism. An introduction to the first leg and some of its applications constitute the curriculum of a typical first year graduate course. There have been all too few books addressed to students who have completed such an introduction, and the present volume is the first such guide in the subject of Cobordism since Robert Stong’s encyclopedic and influential notes of a generation ago. The pioneering work of Pontryagin and Thom forged a deep connection between certain geometric problems (such as the classification of manifolds) and homotopy theory, through the medium of the Thom space. Computations become possible upon stabilization, and this provided some of the first and most compelling examples of “spectra.” Since its inception the subject has thus represented a merger of the Russian and Western mathematical schools. This international tradition was continued with the more or less simultaneous work by Novikov and Milnor on complex cobordism, and later by Quillen. More recently Dennis Sullivan opened the way to the study of “manifolds with singularities,” a study taken up most forcefully by the Russian school, notably by Vershinin, Botvinnik, and Rudyak. Attention to pedagogy is another Russian tradition which you will find amply fulfilled in this book. There is a fine introduction to the stable homotopy category. The subtle and increasingly important issue of phantom maps is addressed here with care. Equally careful is the treatment of orientability, a subject to which the author has
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