New Topological Invariants for Real- and Angle-Valued Maps: An Alternative to Morse-Novikov Theory

254 Pages·2017·5.51 MB·English

by  Dan Burghelea

New Topological
Invariants for Real- and
Angle-Valued Maps
An Alternative to Morse–Novikov Theory
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New Topological
Invariants for Real- and
Angle-Valued Maps
An Alternative to Morse–Novikov Theory
Dan Burghelea
Ohio State University, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Burghelea, Dan, author.
Title: New topological invariants for real- and angle-valued maps :
an alternative to Morse-Novikov theory / by Dan Burghelea (Ohio State University, USA).
Description: New Jersey : World Scientific, 2017. |
Includes bibliographical references and index.
Identifiers: LCCN 2017030717 | ISBN 9789814618243 (hardcover : alk. paper)
Subjects: LCSH: Manifolds (Mathematics) | Mappings (Mathematics) | Topological spaces.
Classification: LCC QA613.2 .B87 2017 | DDC 514/.34--dc23
LC record available at https://lccn.loc.gov/2017030717
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
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Printed in Singapore
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To Ani and Gabriela
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Acknowledgments
ThisbookwasfinalizedduringavisitatMaxPlanckInstituteforMath-
ematics (Bonn, Germany) and at Mathematical Institute of the Romanian
Academy (Bucharest, Romania) (2016). I thank both institutions for their
financial support. I am thankful to colleagues who were attending lectures
on various parts of the material helping to bring improvements to this text
andinparticulartoStefanHaller,coauthorofmanyoftheresultspresented
inthiswork, forsomecriticalobservations. IalsothankAnaBurgheleafor
technical support and Ms. Kwong Lai Fun, editor for World Scientific, for
her patience in working with a disorganized author and help in editing the
manuscript in the present book format.
My warmest thanks go to the copy editor (who does not want his name
mentioned) for helping to convert this work into a hopefully readable text;
withouthishelpthefinalversionofthisbookwouldhavebeenconsiderably
delayed.
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Preface
This book proposes an alternative to Morse-Novikov theory referred
belowasAMN-theoryandisprimarilyaboutthenewtopologicalinvariants
for real- and angle-valued maps which appear in the context of this theory.
Theinvariantsdiscussedareononesideanaloguesofrestpoints,instan-
tons between rest points and closed trajectories of vector fields and on the
other side refine basic topological invariants like Betti numbers, homology
and monodromy.
They are associated to tame maps defined on compact locally con-
tractiblemetricspaces,considerablymoregeneralthanMorsemapsdefined
oncompactsmoothmanifolds,arecomputablebycomputerimplementable
algorithms when the map is simplicial and defined on a finite simplicial
complex and have remarkable robustness and topological properties.
They carry information about the rest points and trajectories of some
classes of flows that admit Lyapunov maps relating dynamics to topology
in the same spirit Morse–Novikov theory does.
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