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2014 | OriginalPaper | Chapter

8. Linearization and Transversality

Authors : Michèle Audin, Mihai Damian

Published in: Morse Theory and Floer Homology

Publisher: Springer London

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Abstract

We show that the spaces of solutions of the Floer equation connecting two periodic orbits is, after a perturbation of the Hamiltonian if necessary, a manifold of dimension the difference of the two indices. For this, we use the analogue of Sard’s theorem in infinite dimension. This requires the assumption that the tangent maps are Fredholm operators. We prove this and compute the index of these operators, which is the dimension of the space of solutions.

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previous chapter The Geometry of the Symplectic Group, the Maslov Index

next chapter Floer Homology: Spaces of Trajectories

And we want to work with continuous functions in order to use the regularity and compactness results Proposition 6.5.3 and Theorem 6.5.4.

We therefore assume that p>2.

We have seen that there is one, because the loops x and y are contractible and our manifold satisfies Assumption 6.2.2.

Up to a factor 2: the usual $\overline {\partial}$ is

$$\frac{1}{2}\Bigl(\frac{\partial}{\partial s}+J_0\frac{\partial}{\partial t}\Bigr).$$

The derivation along X _t is actually the derivation $D_{X_{t}}$ given by the Levi-Cività connection; see Section A.5.

See [17, Chapter 2], if necessary.

Once again [17, Chapter 1], if necessary.

This time, [17, Chapter 4].

If necessary, see [17, Chapter 4] for this inequality:

$$\left \Vert f\star g\right \Vert _{L^p}\leq \left \Vert f\right \Vert _{L^1}\cdot \left \Vert g\right \Vert _{L^p}.$$

See [17, Chapter 9], if necessary.

See [17, Chapter 4], if necessary.

This variant of the mean value inequality is recalled in Proposition C.5.6.

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Bourgeois, F., Oancea, A.: Fredholm theory and transversality for the parametrized and for the S ¹-invariant symplectic action. J. Eur. Math. Soc. 12(5), 1181–1229 (2010) MathSciNetMATH

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Title: Linearization and Transversality
Authors: Michèle Audin
Mihai Damian
Publisher: Springer London
Book: Morse Theory and Floer Homology
Print ISBN: 978-1-4471-5495-2

Electronic ISBN: 978-1-4471-5496-9

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-1-4471-5496-9_8

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