This thesis is split up into two parts each revolving around Floer
homology and quantum cohomology of closed monotone symplectic
manifolds. In the first part we consider symplectic manifolds obtained
by symplectic reduction. Our main result is that a quantum version of
an abelianization formula of Martin holds, which relates
the quantum cohomologies of symplectic quotients by a group and by its
maximal torus. Also we show a quantum version of the Leray-Hirsch
theorem for Floer homology of Lagrangian intersections in the
quotient.
The second part is devoted to Floer homology of a pair of monotone
Lagrangian submanifolds in clean intersection. Under these assumptions
the symplectic action functional is degenerated. Nevertheless
Frauenfelder defines a version of Floer
homology, which is in a certain sense an infinite dimensional analogon
of Morse-Bott homology. Via natural filtrations on the chain level we
were able to define two spectral sequences which serve as a tool to
compute Floer homology. We show how these are used to obtain new
intersection results for simply connected Lagrangians in the product
of two complex projective spaces.
The link between both parts is that in the background the same
technical methods are applied; namely the theory of holomorphic strips
with boundary on Lagrangians in clean intersection. Since all our
constructions rely heavily on these methods we also give a detailed
account of this theory although in principle many results are not new
or require only straight forward generalizations.:1. Introduction
2. Overview of the main results
2.1. Abelianization .
2.2. Quantum Leray-Hirsch theorem
2.3. Floer homology of Lagrangians in clean intersection
3. Background
3.1. Symplectic geometry .
3.2. Hamiltonian action functional
3.3. Morse homology .
3.4. Floer homology
4. Asymptotic analysis
4.1. Main statement .
4.2. Mean-value inequality .
4.3. Isoperimetric inequality
4.4. Linear theory
4.5. Proofs
5. Compactness
5.1. Cauchy-Riemann-Floer equation .
5.2. Local convergence .
5.3. Convergence on the ends
5.4. Minimal energy .
5.5. Action, energy and index estimates
6. Fredholm Theory
6.1. Banach manifold .
6.2. Linear theory
7. Transversality
7.1. Setup
7.2. R-dependent structures
7.3. R-invariant structures .
7.4. Regular points .
7.5. Floer’s ε-norm .
8. Gluing
8.1. Setup and main statement
8.2. Pregluing .
8.3. A uniform bounded right inverse
8.4. Quadratic estimate
8.5. Continuity of the gluing map
8.6. Surjectivity of the gluing map
8.7. Degree of the gluing map
8.8. Morse gluing .
9. Orientations
9.1. Preliminaries and notation
9.2. Spin structures and relative spin structures
9.3. Orientation of caps
9.4. Linear theory .
10.Pearl homology
10.1. Overview .
10.2. Pearl trajectories .
10.3. Invariance .
10.4. Spectral sequences
11.Proofs of the main results
11.1. Abelianization Theorem
11.2. Quantum Leray-Hirsch Theorem .
12.Applications
12.1. Quantum cohomology of the complex Grassmannian
12.2. Lagrangian spheres in symplectic quotients
A. Estimates
A.1. Derivative of the exponential map
A.2. Parallel Transport
A.3. Estimates for strips
B. Operators on Hilbert spaces
B.1. Spectral gap
B.2. Flow operator
C. Viterbo index
D. Quotients of principal bundles by maximal tori
D.1. Compact Lie groups
D.2. The cohomology of the quotient of principle bundles by maximal tori
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:15581 |
| Date | 14 December 2016 |
| Creators | Schmäschke, Felix |
| Contributors | Schwarz, Matthias, Frauenfelder, Urs, Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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