Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2004 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Lectures on Morse Homology

verfasst von: Augustin Banyaga, David Hurtubise

Verlag: Springer Netherlands

Buchreihe : Kluwer Texts in the Mathematical Sciences

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

This book is based on the lecture notes from a course we taught at Penn State University during the fall of 2002. The main goal of the course was to give a complete and detailed proof of the Morse Homology Theorem (Theo­ rem 7.4) at a level appropriate for second year graduate students. The course was designed for students who had a basic understanding of singular homol­ ogy, CW-complexes, applications of the existence and uniqueness theorem for O.D.E.s to vector fields on smooth Riemannian manifolds, and Sard's Theo­ rem. We would like to thank the following students for their participation in the course and their help proofreading early versions of this manuscript: James Barton, Shantanu Dave, Svetlana Krat, Viet-Trung Luu, and Chris Saunders. We would especially like to thank Chris Saunders for his dedication and en­ thusiasm concerning this project and the many helpful suggestions he made throughout the development of this text. We would also like to thank Bob Wells for sharing with us his extensive knowledge of CW-complexes, Morse theory, and singular homology. Chapters 3 and 6, in particular, benefited significantly from the many insightful conver­ sations we had with Bob Wells concerning a Morse function and its associated CW-complex.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
A Morse-Smale function f: M → ℝ. on a finite dimensional compact smooth Riemannian manifold (M, g) gives rise to a chain complex (C * (f), ∂ * ), called the Morse-Smale-Witten chain complex (Definition 7.2), whose chains are generated by the critical points of f and whose boundary operator is defined by counting gradient flow lines (with sign).
Augustin Banyaga, David Hurtubise
Chapter 2. The CW-Homology Theorem
Abstract
In this chapter we introduce singular homology, and we prove the CWHomology Theorem. The CW-Homology Theorem (Theorem 2.15) states that the singular homology H * (X,A; Λ) is isomorphic to the homology of the CWchain complex (C * (X, A; A), a), and it gives a formula for computing the boundary operator ∂ * in the CW-chain complex in terms of the degrees of the attaching maps. We also prove some basDic theorems from homotopy theory. In particular, we prove that if A C X is closed and the inclusion A ⊆ X is a cofibration, then Hk(X, A; Λ) Hk(X/A, *; Λ) for all k (Corollary 2.31).
Augustin Banyaga, David Hurtubise
Chapter 3. Basic Morse Theory
Abstract
The main goal of this chapter is to show how to construct a CW-complex that is homotopy equivalent to a given smooth manifold M using some special functions on M called “Morse” functions (Theorem 3.28). The CW-homology of the resulting CW-complex is isomorphic to the singular homology of M by Theorem 2.15, and hence it is independent of the choice of the Morse func­tion used to build the CW-complex. As a consequence we derive the Morse inequalities. The last section of this chapter is an introduction to Morse-Bott functions.
Augustin Banyaga, David Hurtubise
Chapter 4. The Stable/Unstable Manifold Theorem
Abstract
The main goal of this chapter is to prove the Stable/Unstable Manifold Theorem for a Morse Function (Theorem 4.2). To do this, we first show that a non-degenerate critical point of a smooth function f : M → ℝ on a finite dimensional smooth Riemannian manifold (M, g) is a hyperbolic fixed point of the diffeomorphism φ t coming from the gradient flow (for any fixed t ≠ 0). This is accomplished by computing local formulas for ∇f, d∇f| p ,and dφ t | p with respect to the Riemannian metric g on M (Lemmas 4.3, 4.4, and 4.5).
Augustin Banyaga, David Hurtubise
Chapter 5. Basic Differential Topology
Abstract
In this chapter we prove some results on transversality, general position, orientations, and intersection numbers that will be used in later chapters, including the Inverse Image Theorem (Theorem 5.11) and the Homotopy Transversality Theorems (Theorem 5.17 and Theorem 5.19). As an application of these results we show that the class of Morse functions on a finite dimensional smooth manifold M is locally stable (Corollary 5.24) and dense as a subspace of the space of all smooth functions on M with the uniform topology (Theorem 5.27). We also show that the set of Morse functions on M is an open and dense subspace of the space of all smooth functions on M with the smooth topology (Theorem 5.31). In the last two sections of this chapter we define orientations and intersection numbers, and we prove the Lefschetz Fixed Point Theorem (Theorem 5.50).
Augustin Banyaga, David Hurtubise
Chapter 6. Morse-Smale Functions
Abstract
In this chapter we introduce the Morse-Smale transversality condition for gradient vector fields, and we prove the Kupka-Smale Theorem (Theorem 6.6) which says that the space of smooth Morse-Smale gradient vector fields is a dense subspace of the space of all smooth gradient vector fields on a finite dimensional compact smooth Riemannian manifold (M,g) [92] [135]. We also prove Palis’ λ-Lemma (Theorem 6.17) following [114] and [143], and we derive several important consequences of the λ-Lemma. These consequences include transitivity for Morse-Smale gradient flows (Corollary 6.21), a description of the closure of the stable and unstable manifolds of a Morse-Smale gradient flow (Corollary 6.27), and the fact that for any Morse-Smale gradient flow there are only finitely many gradient flow lines between critical points of relative index one (Corollary 6.29). In the last section of this chapter we present a couple of results due to Franks [59] that relate the stable and unstable manifolds of a Morse-Smale function f: M → ℝ to the cells and attaching maps in the CW-complex X determined by f (see Theorem 3.28).
Augustin Banyaga, David Hurtubise
Chapter 7. The Morse Homology Theorem
Abstract
In this chapter we construct the Morse-Smale-Witten chain complex and prove that its homology coincides with the singular homology. For an interesting history of this complex we refer the reader to Bott’s colorful paper [26]. The story started with a Comptes Rendus Note of the French Academy of Sciences by René Thom in 1949 [145] and culminated with Witten’s paper in 1982 [153], where the boundary operator described here was explicitly written. The way Witten arrived at this boundary operator was through super-symmetric mechanics. Before, Morse and Smale’s work had already found the ideas required to make rigorous Witten’s “physicist’s” proof: these ideas have been explained in the preceeding chapters. This is why we call the resulting complex the Thom-Morse-Smale-Witten chain complex, or simply, the MorseSmale-Witten chain complex. In the literature it is sometimes called the Witten complex [26].
Augustin Banyaga, David Hurtubise
Chapter 8. Morse Theory On Grassmann Manifolds
Abstract
In this chapter we show how Bott’s perfect Morse functions (discussed in Example 3.7) are examples of a more general class of Morse-Smale functions defined on the complex Grassmann manifolds. The Morse-Smale functions, f A : G n,n+k (ℂ) → ℝ, are defined analogous to the Morse functions constructed in Theorem 3.8.
Augustin Banyaga, David Hurtubise
Chapter 9. An Overview of Floer Homology Theories
Abstract
Floer homology theories are attempts to build in infinite dimensions the equivalent of the Morse-Smale-Witten chain complex (C * (f ), ∂ * ) The finite dimensional manifold M is replaced by an infinite dimensional manifold M and the Morse-Smale function f : M → ℝ is replaced by some ”functional” on M.
Augustin Banyaga, David Hurtubise
Backmatter
Metadaten
Titel
Lectures on Morse Homology
verfasst von
Augustin Banyaga
David Hurtubise
Copyright-Jahr
2004
Verlag
Springer Netherlands
Electronic ISBN
978-1-4020-2696-6
Print ISBN
978-90-481-6705-0
DOI
https://doi.org/10.1007/978-1-4020-2696-6