##
**Morse theory and Floer homology.
(Théorie de Morse et homologie de Floer.)**
*(French)*
Zbl 1217.57001

Savoirs Actuels. Les Ulis: EDP Sciences; Paris: CNRS Éditions (ISBN 978-2-7598-0518-1/pbk; 978-2-271-07087-6/pbk). 548 p. (2010).

The objective of this book is to provide an (almost) self-contained presentation of Floer homology.

Morse theory was developed during the 60ies. The idea was to recover information about any compact manifold \(Y\) from the analytic data of a given generic continuous map \(f\) from \(Y\) to \(\mathbb R\). It led to the notion of Morse homology which is a homological invariant of \(Y\) obtained by counting flowlines between critical points of \(f\). In [J. Differ. Geom. 28, No.3, 513–547 (1988; Zbl 0674.57027)], A. Floer adapted the construction to fit the study of some infinite-dimensional spaces. Using his invariant, he solved a conjecture of Arnold saying that the number of periodic solutions of a Hamiltonian system over a compact symplectic manifold \(W\) is bounded below by topological properties of \(W\), namely the total dimension of its Morse homology.

This book provides a complete definition of Floer homology. As an application, it gives a proof of the Arnold conjecture. As an introduction to the subject, a first part is dedicated to the definition of Morse homology. The book in written in French and is addressed to graduate students or non specialist researchers looking for some detailed presentation of Floer homology. It contains many uncorrected exercises.

The first part is devoted to Morse theory for compact submanifolds of \(\mathbb R^n\), possibly with boundary. Most of the substance follows W. J. Milnor’s books [Morse theory. Based on lecture notes by M. Spivak and R. Wells, Princeton, N.J.: Princeton University Press (1963; Zbl 0108.10401) and Lectures on the \(h\)-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, N.J.: Princeton University Press (1965; Zbl 0161.20302)]. The way Morse theory is treated is focused on Morse homology, with no digression. The first chapter defines and proves the existence of Morse functions. The second chapter deals with gradient-like vector fields which enable the definition of stable and unstable manifolds. The way level hypersurfaces deform is also discussed there. It ends with the classification of compact one-dimensional manifolds. The third chapter defines Morse complexes and Morse homologies with \(\mathbb Z\)-coefficients using a trajectory-space point of view. The fourth chapter concludes the first part by giving some properties of Morse homology – e.g. Künneth formula, Poincaré duality, functoriality, exact sequences – and some applications – e.g. Morse inequalities, Brouwer theorem, Borsuk-Ulam theorem. Every chapter ends with some uncorrected exercises.

With the first part as a guideline, the second part builds Floer homology for symplectic manifolds which are symplectically aspherical and such that the symplectic tangent bundle is trivial on 2-spheres. Most chapters begin with an overview which is sufficient to understand the ins and outs as well as the main steps of the reasoning. All the proofs are then given. Since they are written in detail, no familiarity with the techniques involved is required. This part is structured as follows. The fifth chapter reviews, with proofs, the necessary background of symplectic geometry. After stating the Arnold conjecture, the sixth chapter sets the main objects of the construction. First an action functional \({\mathcal A}\) is given, whose critical points correspond to periodic solutions of a given Hamiltonian system. Then a metric is defined, deduced from the choice of an almost-complex structure, and inducing a gradient vector field \(\xi\) for \({\mathcal A}\). Finally the so-called Floer equation is stated, whose solutions correspond to trajectories for \(\xi\). All the needed properties of these objects, such as compactness, are proven there. Note that section 6.2 is a one-page summary of the strategy developed in the book. Chapter 7 associates a Maslov index to every critical point, enhancing the set of critical points with a grading. Chapter 8 is the analytic core of the book. It deals with transversality, and proves that the spaces of trajectories between two critical points are generically smooth manifolds of finite dimension. These dimensions are computed in terms of Maslov indices. The Floer complex with \(\mathbb Z/\mathbb Z2\)-coefficients is finally defined in chapter 9. It is generated by critical points of \({\mathcal A}\) and the differential counts trajectories between generators with adjacent Maslov gradings. The issue is essentially to glue trajectories so that the classical Morse argument can be used to prove that the differential is actually a differential. The tenth chapter relates the Floer and Morse complexes by giving some particular Hamiltonian systems for which they coincide. This is a key point for proving the Arnold conjecture. Chapter 11 proves that Floer homology is well defined in the sense that it does not depend on the choice of the Hamiltonian system nor on the choice of the almost complex structure. Chapters 12 and 13 summarize some technical analytical proofs about, amongst others, elliptical regularity. Uncorrected exercises for the different chapters are gathered at the end of the second part.

A three chapter appendix introduces, with only few proofs, the prerequisite material from differential geometry, topological algebra and analysis.

Morse theory was developed during the 60ies. The idea was to recover information about any compact manifold \(Y\) from the analytic data of a given generic continuous map \(f\) from \(Y\) to \(\mathbb R\). It led to the notion of Morse homology which is a homological invariant of \(Y\) obtained by counting flowlines between critical points of \(f\). In [J. Differ. Geom. 28, No.3, 513–547 (1988; Zbl 0674.57027)], A. Floer adapted the construction to fit the study of some infinite-dimensional spaces. Using his invariant, he solved a conjecture of Arnold saying that the number of periodic solutions of a Hamiltonian system over a compact symplectic manifold \(W\) is bounded below by topological properties of \(W\), namely the total dimension of its Morse homology.

This book provides a complete definition of Floer homology. As an application, it gives a proof of the Arnold conjecture. As an introduction to the subject, a first part is dedicated to the definition of Morse homology. The book in written in French and is addressed to graduate students or non specialist researchers looking for some detailed presentation of Floer homology. It contains many uncorrected exercises.

The first part is devoted to Morse theory for compact submanifolds of \(\mathbb R^n\), possibly with boundary. Most of the substance follows W. J. Milnor’s books [Morse theory. Based on lecture notes by M. Spivak and R. Wells, Princeton, N.J.: Princeton University Press (1963; Zbl 0108.10401) and Lectures on the \(h\)-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, N.J.: Princeton University Press (1965; Zbl 0161.20302)]. The way Morse theory is treated is focused on Morse homology, with no digression. The first chapter defines and proves the existence of Morse functions. The second chapter deals with gradient-like vector fields which enable the definition of stable and unstable manifolds. The way level hypersurfaces deform is also discussed there. It ends with the classification of compact one-dimensional manifolds. The third chapter defines Morse complexes and Morse homologies with \(\mathbb Z\)-coefficients using a trajectory-space point of view. The fourth chapter concludes the first part by giving some properties of Morse homology – e.g. Künneth formula, Poincaré duality, functoriality, exact sequences – and some applications – e.g. Morse inequalities, Brouwer theorem, Borsuk-Ulam theorem. Every chapter ends with some uncorrected exercises.

With the first part as a guideline, the second part builds Floer homology for symplectic manifolds which are symplectically aspherical and such that the symplectic tangent bundle is trivial on 2-spheres. Most chapters begin with an overview which is sufficient to understand the ins and outs as well as the main steps of the reasoning. All the proofs are then given. Since they are written in detail, no familiarity with the techniques involved is required. This part is structured as follows. The fifth chapter reviews, with proofs, the necessary background of symplectic geometry. After stating the Arnold conjecture, the sixth chapter sets the main objects of the construction. First an action functional \({\mathcal A}\) is given, whose critical points correspond to periodic solutions of a given Hamiltonian system. Then a metric is defined, deduced from the choice of an almost-complex structure, and inducing a gradient vector field \(\xi\) for \({\mathcal A}\). Finally the so-called Floer equation is stated, whose solutions correspond to trajectories for \(\xi\). All the needed properties of these objects, such as compactness, are proven there. Note that section 6.2 is a one-page summary of the strategy developed in the book. Chapter 7 associates a Maslov index to every critical point, enhancing the set of critical points with a grading. Chapter 8 is the analytic core of the book. It deals with transversality, and proves that the spaces of trajectories between two critical points are generically smooth manifolds of finite dimension. These dimensions are computed in terms of Maslov indices. The Floer complex with \(\mathbb Z/\mathbb Z2\)-coefficients is finally defined in chapter 9. It is generated by critical points of \({\mathcal A}\) and the differential counts trajectories between generators with adjacent Maslov gradings. The issue is essentially to glue trajectories so that the classical Morse argument can be used to prove that the differential is actually a differential. The tenth chapter relates the Floer and Morse complexes by giving some particular Hamiltonian systems for which they coincide. This is a key point for proving the Arnold conjecture. Chapter 11 proves that Floer homology is well defined in the sense that it does not depend on the choice of the Hamiltonian system nor on the choice of the almost complex structure. Chapters 12 and 13 summarize some technical analytical proofs about, amongst others, elliptical regularity. Uncorrected exercises for the different chapters are gathered at the end of the second part.

A three chapter appendix introduces, with only few proofs, the prerequisite material from differential geometry, topological algebra and analysis.

Reviewer: Benjamin Audoux (Marseille)