Moduli Spaces of Flowlines with Cascades and Morse-Bott Homology
- 발행기관 서강대학교 일반대학원
- 지도교수 김준태
- 발행년도 2026
- 학위수여년월 2026. 2
- 학위명 석사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/sogang/000000082573
- UCI I804:11029-000000082573
- 본문언어 영어
- 저작권 논문은 저작권에 의해 보호받습니다.
초록(요약문)
Morse theory provides a geometric framework relating the topology of a smooth manifold to the critical points of a non-degenerate smooth function on the manifold. Its generalization, Morse–Bott theory, relaxes the non-degeneracy condition on smooth functions, allowing critical submanifolds. This thesis provides a self-contained construction of Morse–Bott homology based on the geometric analysis of flowlines. We begin by reviewing Morse homology, analyzing the moduli spaces of negative gradient flowlines. In particular, we establish transversality and compactness results for these moduli spaces and verify that the resulting homology is a well-defined invariant of closed smooth manifolds. Next, we develop the framework for Morse–Bott homology. We introduce flowlines with cascades and investigate the corresponding transversality and compactness adapted to this setting. Finally, we show that Morse–Bott homology is isomorphic to Morse homology.
more목차
Abstract iii
1 Introduction 1
2 Moduli Spaces of Morse Flowlines 4
2.1 Morse Flowlines and Energy 4
2.2 Transversality 9
2.2.1 Transversality for Submanifolds 10
2.2.2 Properties of Transversality 13
2.2.3 Fredholm Theory and Transversality for Banach Bundles 19
2.2.4 Transversality for Moduli Spaces 23
2.3 Compactness 31
2.3.1 Convergence of Morse Flowlines 31
2.3.2 Convergence to Broken Flowlines 38
2.4 Gluing 41
3 Morse Homology 43
3.1 Definition of Morse Homology 43
3.2 Examples of Morse Homology 44
3.3 Invariance Theorem 49
3.3.1 Continuation Solutions and Chain Maps 50
3.3.2 Homotopies of Homotopies and Chain Homotopy 52
3.3.3 Proof of the Main Theorem 55
4 Moduli Spaces of Flowlines with Cascades 58
4.1 Morse–Bott Setup 58
4.2 Preliminary Observations 61
4.3 Manifold Structures of Moduli Spaces 64
4.3.1 Manifolds with Corners 64
4.3.2 Transversality 65
4.4 Gluing and Compactness 75
5 Morse–Bott Homology 79
5.1 Definition of Morse–Bott Homology 79
5.2 An Example of Morse–Bott Homology 80
5.3 Continuation Solutions and Invariance 82
5.4 Morse Homology and Morse–Bott Homology 85
A Hessian Operator 86
Bibliography 88
