Morse Theory and Floer Homology (Universitext)
Michèle Audin, Mihai Damian
Language: English
Pages: 596
ISBN: 1447154959
Format: PDF / Kindle (mobi) / ePub
This book is an introduction to modern methods of symplectic topology. It is devoted to explaining the solution of an important problem originating from classical mechanics: the 'Arnold conjecture', which asserts that the number of 1-periodic trajectories of a non-degenerate Hamiltonian system is bounded below by the dimension of the homology of the underlying manifold.
The first part is a thorough introduction to Morse theory, a fundamental tool of differential topology. It defines the Morse complex and the Morse homology, and develops some of their applications.
Morse homology also serves a simple model for Floer homology, which is covered in the second part. Floer homology is an infinite-dimensional analogue of Morse homology. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the Arnold conjecture. The building blocks of Floer homology are more intricate and imply the use of more sophisticated analytical methods, all of which are explained in this second part.
The three appendices present a few prerequisites in differential geometry, algebraic topology and analysis.
The book originated in a graduate course given at Strasbourg University, and contains a large range of figures and exercises. Morse Theory and Floer Homology will be particularly helpful for graduate and postgraduate students.
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We deduce from this that ΦG ◦ ΦF and ΦH induce the same morphism in the homology. 3.5 Cobordisms Let us now consider cobordisms. A cobordism is a manifold with boundary V whose boundary ∂V is decomposed into ∂V = ∂− V ∪ ∂+ V (the two parts ∂± V are unions of arbitrarily chosen components of ∂V ). We generalize the construction of Section 2.3.a. We fix a vector field X (constructed, for example, using a partition of unity) on a neighborhood of ∂V in V that is: • pointing outward along ∂+ V •
groups. 82 4 Morse Homology, Applications Proposition 4.2.1. The map Φ defines an isomorphism of complexes (C (f ) ⊗ C (g), ∂X ⊗ 1 + 1 ⊗ ∂Y ) −→ (C (f + g), ∂(X,Y ) ). Proof. Let a be a critical point of index i of f and let a be a critical point of index j of g. On the one hand, we have Φ(∂X ⊗ 1 + 1 ⊗ ∂Y )(a ⊗ a ) = Φ(∂X (a) ⊗ a + a ⊗ ∂Y (a )) nX (a, b)b ⊗ a + =Φ = nY (a , b )a ⊗ b b ∈Critj−1 (g) b∈Criti−1 (f ) nX (a, b)(b, a ) + nY (a , b )(a, b ), b ∈Critj−1 (g) b∈Criti−1 (f ) and on
IRMA Université Louis Pasteur Strasbourg Cedex, France Translation from the French language edition: Théorie de Morse et homologie de Floer by Michèle Audin and Mihai Damian EDP Sciences ISBN 978-2-7598-0704-8 Copyright © 2010 EDP Sciences, CNRS Editions, France. http://www.edpsciences.org/ http://www.cnrseditions.fr/ All rights reserved ISSN 0172-5939 e-ISSN 2191-6675 Universitext ISBN 978-1-4471-5495-2 e-ISBN 978-1-4471-5496-9 DOI 10.1007/978-1-4471-5496-9 Springer London Heidelberg New York
induction on s: • We have P1,1 because ω(X, Y ) = ω(AX, AY ) = λμω(X, Y ). • Next, we assume that P1,s is true. Let X be an eigenvector of A and let Y ∈ Ker(A − μ Id)s+1 . We have ω(X, Y ) = ω(AX, AY ) = λω(X, (A − μ Id)Y ) + λμω(X, Y ) = λμω(X, Y ) because (A − μ Id)Y ∈ Ker(A − μ Id)s . In the same manner, we have property Pr,1 . To show Pr,s by induction, we verify that Pr,s+1 and Pr+1,s =⇒ Pr+1,s+1 . Therefore, let X ∈ Ker(A − λ Id)r+1 and Y ∈ Ker(A − μ Id)s+1 . We have ω(X, Y ) = ω(AX, AY ) =
the remaining n − k variables whose partial derivatives of orders 1 and 2 are all zero at the point in question. 1.3 The Morse Lemma, the Index of a Critical Point 13 We will prove the result by induction on the dimension. We begin with the case n = 1, where 1 f (x) = f (0) + f (0)x2 + ε(x)x2 2 = f (0) ± ax2 (1 + ε(x)) for a positive real number a and a C∞ function ε, namely ε(x) = 1 2 x f (3) (t)(x − t)2 dt. 0 We set x1 = ϕ(x) = x a(1 + ε(x)). √ It is clear that ϕ is a local
