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  • Textbook
  • © 2007

An Invitation to Morse Theory

  • Provides a useful introduction to Morse Theory

  • Covers many of the most important topics in Morse theory, along with applications

  • Contains many very good exercises

  • Far more up-to-date and less specialized than any other book on Morse Theory

Part of the book series: Universitext (UTX)

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eBook USD 44.99
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  • ISBN: 978-0-387-49510-1
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Table of contents (5 chapters)

  1. Front Matter

    Pages I-XIV
  2. Morse Functions

    Pages 1-22
  3. Applications

    Pages 87-150
  4. Exercises and Solutions

    Pages 193-232
  5. Back Matter

    Pages 233-241

About this book

This self-contained treatment of Morse Theory focuses on applications and is intended for a graduate course on differential or algebraic topology. The book is divided into three conceptually distinct parts. The first part contains the foundations of Morse theory (over the reals). The second part consists of applications of Morse theory over the reals, while the last part describes the basics and some applications of complex Morse theory, a.k.a. Picard-Lefschetz theory.

This is the first textbook to include topics such as Morse-Smale flows, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The exposition is enhanced with examples, problems, and illustrations, and will be of interest to graduate students as well as researchers. The reader is expected to have some familiarity with cohomology theory and with the differential and integral calculus on smooth manifolds.

Liviu Nicolaescu is Associate Professor of Mathematics at University of Notre Dame.

Keywords

  • Algebraic topology
  • Topology
  • algebra
  • calculus
  • cohomology
  • cohomology theory
  • function
  • homology
  • manifold
  • mathematics

Reviews

From the reviews:

"Morse theory, a tool within differential topology, strategically studies a given abstract smooth manifold by first imposing on it a nearly arbitrary numerical function, and then cleverly extracting from it purely topological information. … Undergraduates will see that the foundations of this advanced topic build directly on a(n honest) course in multivariable calculus … . Primarily for mathematics students. Summing Up: Recommended. Upper-division undergraduates through professionals." (D. V. Feldman, CHOICE, Vol. 45 (6), February, 2008)

"The book is a nicely written self-contained introduction to Morse theory … . will be useful for mathematicians of various levels, including graduate students and researchers." (Michael Farber, Zentralblatt MATH, Vol. 1131, 2008)

"Nicolaescu’s book starts with the basics of Morse theory over the reals … . The discussion continuously presents some really nice and well chosen applications of the theory, and finally lets the reader see, that the whole theory can go on to complex, where the set of regular values, that is disconnected by nature over the reals, becomes connected. … This book is warmly recommended for interested graduate students and researcher … ." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

"Nicolaescu’s book complements previous books on Morse theory by quickly developing the foundations of the subject in terms of gradient-like vector fields and discussing applications not found in other books on Morse theory. … the book is recommended for graduate students and researchers … ." (David E. Hurtubise, Mathematical Reviews, Issue 2009 m)

“This book offers a beautiful self contained introduction to Morse theory. It should be accessible to graduate students familiar with the basics of algebraic topology and differential geometry. … it provides a complete proof of the Picard–Lefschetz formula. The book concludes with a number of exercises and solutions to some of them.” (St. Haller, Monatshefte für Mathematik, Vol. 157 (1), May, 2009)

Authors and Affiliations

  • Mathematics Department, University of Notre Dame, Notre Dame, USA

    Liviu Nicolaescu

Bibliographic Information

Buying options

eBook USD 44.99
Price excludes VAT (USA)
  • ISBN: 978-0-387-49510-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout