Lectures on Morse Homology (Texts in the Mathematical Sciences, 29) 2004th Edition by Augustin Banyaga (PDF)

Description

This book offers a detailed presentation of results needed to prove the Morse Homology Theorem using classical techniques from algebraic topology and homotopy theory. The text presents results that were formerly scattered in the mathematical literature, in a single reference with complete and detailed proofs. The core material includes CW-complexes, Morse theory, hyperbolic dynamical systems (the Lamba-Lemma, the Stable/Unstable Manifold Theorem), transversality theory, the Morse-Smale-Witten boundary operator, and Conley index theory.

User’s Reviews

Editorial Reviews: Review From the reviews of the first edition:”This book presents in great detail all the results one needs to prove the Morse homology theorem using classical techniques from algebraic topology and homotopy theory. … This book collects all these results together into a single reference with complete and detailed proofs. … With the stress on completeness and by its elementary approach to Morse homology, this book is suitable as a textbook for a graduate level course, or as a reference for working mathematicians and physicists.” (Bulletin Bibliographique, Vol. 51 (1-2), 2005)”This book provides a treatment of finite-dimensional Morse theory and its associated chain complex, pitched at a level appropriate to early-stage graduate students. … Throughout, the authors take pains to make the material accessible, and … extensive references are provided. … Many well-drawn figures are provided to clarify the text, and there are over 200 exercises, with hints for some of them in the back. … Banyaga and Hurtubise’s book provides a valuable service by introducing young mathematicians to a circle of ideas … .” (Michael J. Usher, Mathematical Reviews, Issue 2006 i)”This book is an exposition of the ‘classical’ approach to finite dimensional Morse homology. … This book presents in great detail all the results one needs to prove the Morse Homology theorem … . References to the literature are provided throughout the book … . A lot of examples, suggestive figures and diagrams in every chapter and many useful exercises at the end of the chapters makes this book a good and attractive textbook (as well as an excellent monograph). … The bibliography is exhaustive.” (Ioan Pop, Zentralblatt MATH, Vol. 1080, 2006)

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Morse theory (the study of the critical points of a nondegenerate smooth function on a smooth manifold and in particular their topological implications) is a classic subject whose standard texts, Milnor’s

⭐and

⭐and Morse’s

⭐, date from the ’60s or earlier. Starting in the late ’80s, due to the work of Floer and Witten, a new perspective emerged, both on the old finite-dimensional theory (Witten’s rediscovery of Morse homology) and on infinite-dimensional Morse theory, initiated by Floer, which resembles gauge theory and has deep ties to symplectic geometry. This book by Banyaga and Hurtubise attempts to, first of all, present the classic finite-dimensional theory but in modern dynamical-systems terminology, and also give a brief introduction to the newer developments in Floer homology. It is more successful on accomplishing the former, as its treatment of Floer homology is rather cursory. It is largely self-contained, as an introductory graduate text, requiring only a knowledge of singular homology theory and basic topology and analysis.The book includes 2 chapters on prerequisite material, namely, CW homology and basic differential topology, but if you don’t know these subjects already, in particular, the differential topology, this won’t be enough to help you. All the standard results in Morse theory, as can be found in Milnor’s books or Matsumoto’s

⭐, are presented, although some of the proofs are given in more general form, such as Palais’s proof of the fundamental Morse lemma, which also works on Banach spaces. There are many topics that have been collected from the literature, including the full set of Morse inequalities, such as the polynomial Morse inequalities, and Morse-Bott functions, which usually don’t appear in other books on Morse theory. There’s also an extensive treatment of dynamical systems, which are implicit in the Morse theory but not usually formally addressed, including a modestly analytically challenging proof of the stable/unstable manifold theorem, which forms the core of the book, as well the lambda-lemma, Kupka-Smale theorem, which shows the Morse-Smale gradient vector fields are generic, and the Conley index. The culmination of these technical results is the Morse homology theorem, which relates the CW homology (hence singular homology) of a compact smooth Riemannian manifold to the homology of a Morse-Smale-Witten chain complex, which is particularly easy to compute, only involving counting flow lines between critical points with signs coming from orientations and flow directions.The last 2 chapters are more specialized and can be omitted in a short course or on a first reading. There’s one on the Morse theory of Grassmann manifolds by constructing perfect Morse-Bott functions on them. Here previous knowledge of Lie groups and algebras is necessary, although some of it is derived. A large part of the proof of the existence of such Mose-Bott functions is practically copied verbatim from Milnor’s (“Morse Theory”) proof that Morse functions are generic. The payoff in this chapter for so much effort is somewhat modest – it ends by only stating some similar results, such as the cohomology of the classification space for complex bundles and Morse theory for Lie groups, that should probably have been proved here. The last chapter and a half of the book is primarily concerned with applications to symplectic geometry, including Floer homology. A ton of material is summarized very quickly, often by just copying sections of several papers, in a manner that will not be of much help for beginners. The main point of this seems to be showing how Morse theory can be applied or extended, or even just showing THAT it can be applied.This is a textbook, so there is a wealth of exercises and a fair amount of examples. For most of the book, the proofs develop very slowly, with all of the steps explicitly spelled out, making them easy to follow. The emphasis on using modern dynamical systems terminology and theorems, the presentation of multiple proofs of the same theorem, practical demonstrations on how Morse homology can be computed, and the inclusion of more obscure results are also pluses, as is the proof of the Morse homology theorem, which differs from other proofs in that it doesn’t involve infinite-dimensional techniques or require overly restrictive assumptions on the Morse function (such as it being self-indexing or compatible with a Riemann metric). It is unusual among books on Morse theory in that it is both modern (unlike Milnor, Morse, or Matsumoto) and classical (unlike Schwarz’s

⭐or Chang’s

⭐). The main drawbacks of the book are that it doesn’t do a good job of explaining or using the Floer theory and homology on Grassmann manifolds presented at the end of the book, a large amount of material seems to be copied from other sources, it takes a long time to arrive at the most interesting results (certainly longer than necessary), and there is a plethora of mathematical typos, much more than one normally finds (and the bibliography includes 2 of the most bizarre kinds of typos I’ve ever seen).

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