Equivariant Ordinary Homology and Cohomology
Filling a gap in the literature, this book takes the reader to the frontiers of equivariant topology, the study of objects with specified symmetries. The discussion is motivated by reference to a list of instructive “toy” examples and calculations in what
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Steven R. Costenoble Stefan Waner
Equivariant Ordinary Homology and Cohomology
Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Michel Brion, Grenoble Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg
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More information about this series at http://www.springer.com/series/304
Steven R. Costenoble • Stefan Waner
Equivariant Ordinary Homology and Cohomology
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Stefan Waner Department of Mathematics Hofstra University Hempstead New York, USA
Steven R. Costenoble Department of Mathematics Hofstra University Hempstead New York, USA
ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-50447-6 DOI 10.1007/978-3-319-50448-3
ISSN 1617-9692 (electronic) ISBN 978-3-319-50448-3 (eBook)
Library of Congress Control Number: 2016963346 Mathematics Subject Classification (2010): Primary 55N91, Secondary 55P91, 57R91, 55N25, 55P42, 55P20, 55R70, 55R91 © Springer International Publishing AG 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Introduction
Poincaré duality in ordinary homology is a powerful tool in the study of manifolds. In the presence of a smooth action of a compact Lie group, it has not been clear what the appropriate analogue is or whether there even is one. The fundamental question that must be addressed is what one means by equivariant ordinary homology and cohomology. Historically, one of the earliest candidates was Borel homology, which, for a Gspace X, is the nonequivariant homology of EGG X, where EG is a nonequivariantly contractible free G-space [3]. This is still what many people mea
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