Publications
Detailed Information
Morse-Bott Spectral Sequences and the Links of Singularities : 모스-보트 스펙트럴 수열과 특이점의 고리
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | Otto van Koert | - |
| dc.contributor.author | 권명기 | - |
| dc.date.accessioned | 2017-07-14T00:42:57Z | - |
| dc.date.available | 2017-07-14T00:42:57Z | - |
| dc.date.issued | 2017-02 | - |
| dc.identifier.other | 000000141031 | - |
| dc.identifier.uri | https://hdl.handle.net/10371/121321 | - |
| dc.description | 학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2017. 2. Otto van Koert. | - |
| dc.description.abstract | In this thesis, we construct spectral sequences converging to symplectic homology and equivariant symplectic homology groups. We use Morse-Bott type Hamiltonians and a natural action filtration. Those spectral sequences are called Morse-Bott spectral sequences. We apply the spectral sequences to a certain kind of symplectic manifolds with boundary, namely Milnor fibers whose boundaries are the links of singularities. In special cases, such as links of weighted homogeneous polynomials, they admit a nice symmetry along a periodic Reeb flow of a contact form. By means of those special symmetric feature, we present a systematic way of computing equivariant symplectic homology groups and its mean Euler characteristic. We obtain several applications of these machineries to exotic contact structures. | - |
| dc.description.tableofcontents | 1 Introduction 1
1.1 Contact structures on the links of singularities 1 1.2 Morse-Bott spectral sequence 3 1.3 Applications 5 2. Preliminaries 8 2.1 Symplectic manifolds 8 2.1.1 Hamiltonians 8 2.1.2 Almost complex structures 9 2.2 Contact manifolds 10 2.2.1 Reeb vector fields 10 2.2.2 Symplectizations 11 2.3 Symplectic fillings of contact manifolds 12 3. Symplectic Homology 15 3.1 Completion 15 3.2 Admissible Hamiltonians 16 3.3 Conley-Zehnder index 19 3.3.1 For a path of symplectic matrices 19 3.3.2 Robbin-Salamon index 21 3.3.3 For periodic orbits 24 3.3.4 Morse index and Conley-Zehnder index 25 3.3.5 Linearized Hamiltonian flow and Reeb flow 26 3.4 Moduli spaces of Floer trajectories 27 3.4.1 Admissible almost complex structures 28 3.4.2 Transversality 29 3.4.3 A priori energy bound 31 3.4.4 Maximum principle 32 3.4.5 Bubbling phenomenon 34 3.4.6 Floer trajectories of small Hamiltonians 35 3.5 Definition of symplectic homology 37 3.5.1 Hamiltonian Floer homology 37 3.5.2 Continuation homomorphisms 38 3.5.3 A direct system of Hamiltonians and symplectic homology groups 40 3.5.4 A natural action filtration 42 3.6 Examples 44 3.6.1 Ball 44 3.6.2 Annulus 44 3.6.3 Cotangent bundles 45 3.7 Invariance 45 3.7.1 Liouville isomorphisms 46 3.7.2 Liouville homotopies 47 4. Equivariant Symplectic Homology 48 4.1 S 1 -equivariant Morse homology 48 4.1.1 S 1 -equivariant Morse complex 49 4.1.2 S 1 -equivariant Morse differentials 49 4.2 S 1 -equivariant symplectic homology 52 4.2.1 S 1 -invariant action functional 52 4.2.2 S 1 -equivariant Floer complex 55 5. Morse-Bott Spectral Sequence for Morse Homology 57 5.1 Fredholm operators 57 5.2 Morse-Bott functions 59 5.2.1 Definition 59 5.2.2 Standard Perturbation 60 5.3 Morse Homology with local coefficient systems 61 5.4 Local Morse Homology 69 5.4.1 Local Morse homology of Σ 70 5.4.2 Construction of the local system L Σ 72 5.4.3 Canonical isomorphisms 79 5.5 Morse-Bott spectral sequence for Morse homology 82 6. Morse-Bott spectral sequences for symplectic homology 86 6.1 Morse-Bott type Hamiltonians 86 6.1.1 Standard perturbation 88 6.1.2 A priori energy bound 88 6.2 Local Floer homology 90 6.2.1 Definition 90 6.2.2 Local Floer homology and Morse homology 91 6.2.3 Construction of the local system L Σ 94 6.3 An action filtration of Floer chain complex 98 6.4 Morse-Bott spectral sequences for symplectic homology 100 6.4.1 For Hamiltonian Floer homology 100 6.4.2 For symplectic homology 101 6.4.3 For equivariant symplectic homology 104 7. Links of Singularities 107 7.1 Backgrounds on Stein manifolds 107 7.1.1 Definitions and examples 107 7.1.2 Stein domains 110 7.1.3 Subcritical Stein domains 111 7.2 Links of singularities 113 7.3 A natural Stein filling 114 7.3.1 A smoothed variety 114 7.3.2 Milnor fibers 115 7.3.3 Topology of complex hypersurface and its link 116 7.3.4 The middle dimensional homology group of Σ(f) 117 7.4 Weighted homogeneous polynomials 118 7.4.1 The middle dimensional homology for weighted homogeneous polynomials 119 7.4.2 Randells algorithm: Torsion part of H n−1 (Σ(a)) 122 7.4.3 Brieskorn spheres 123 7.4.4 Equivariant singular homology of Σ(f) 124 7.5 Periodic Reeb flows on the links of weighted homogeneous polynomials 126 7.6 Reeb dynamics of (Σ(f), α) 129 7.6.1 Brieskorn case 130 7.6.2 General weighted homogeneous case 132 7.7 Robbin-Salamon indices 135 7.7.1 Maslov index and the first Chern number: homogeneous case 140 7.7.2 Maslov index and the first Chern number: the weighted homogeneous case 142 8. Equivariant symplectic homology of the links of weighted homogenous polynomials 144 8.1 First examples 144 8.1.1 Example: Σ(3, 2, 2, 2) 145 8.1.2 Example: E 7 -singularity with n=3 151 8.2 Simple singularities 151 8.3 Homogeneous polynomials 152 9. Invariants of contact structures from symplectic homology 155 9.1 Towards invariants of contact structures 155 9.2 The mean Euler characteristic 157 9.2.1 Definition 157 9.2.2 Invariance 158 9.2.3 The mean Euler characteristic and the subcritical handle attachment 162 9.2.4 The case when SH=0 163 9.2.5 Computations via spectral sequences 164 9.2.6 Some explicit formulas 166 9.3 Applications of the mean Euler characteristic 168 Bibliography 174 Abstract (in Korean) 179 | - |
| dc.format | application/pdf | - |
| dc.format.extent | 3722154 bytes | - |
| dc.format.medium | application/pdf | - |
| dc.language.iso | en | - |
| dc.publisher | 서울대학교 대학원 | - |
| dc.subject | Morse-Bott spectral sequence | - |
| dc.subject | link of singularity | - |
| dc.subject | symplectic homology | - |
| dc.subject | Milnor fiber | - |
| dc.subject | mean Euler characteristic | - |
| dc.subject.ddc | 510 | - |
| dc.title | Morse-Bott Spectral Sequences and the Links of Singularities | - |
| dc.title.alternative | 모스-보트 스펙트럴 수열과 특이점의 고리 | - |
| dc.type | Thesis | - |
| dc.description.degree | Doctor | - |
| dc.citation.pages | 178 | - |
| dc.contributor.affiliation | 자연과학대학 수리과학부 | - |
| dc.date.awarded | 2017-02 | - |
- Appears in Collections:
- Files in This Item:
Item View & Download Count
Items in S-Space are protected by copyright, with all rights reserved, unless otherwise indicated.