莫爾斯理論入門 [An Invitation to Morse Theory]

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齣版社: 世界圖書齣版公司
ISBN:9787510027291
版次:1
商品編碼:10762448
包裝:平裝
外文名稱:An Invitation to Morse Theory
開本:24開
齣版時間:2010-09-01
用紙:膠版紙
頁數:241
正文語種:英文

具體描述

內容簡介

As the the title suggests, the goal of this book is to give the reader a taste of the “unreasonable effectiveness” of Morse theory. The main idea behind thistechnique can be easily visualized.
Suppose M is a smooth, compact manifold, which for simplicity we as-sume is embedded in a Euclidean space E. We would like to understand basictopological invariants of M such as its homology, and we attempt a “slicing” technique.

目錄

Preface
Notations and conventions
1 Morse Functions
1.1 The Local Structure of Morse Functions
1.2 Existence of Morse Functions

2 The Topology of Morse Functions
2.1 Surgery,Handle Attachment.and Cobordisms
2.2 The Topology of Sublevel Sets
2.3 Morse Inequalities
2.4 Morse-Smale Dynamics
2.5 Morse-Floer Homology
2.6 Morse-Bott Functions
2.7 Min-Max Theory

3 Applications
3.1 The Cohomology of Complex Grassmannians
3.2 Lefschetz Hyperplane Theorem
3.3 Symplectic Manifolds and Hamiltonian Flows
3.4 Morse Theory of Moment Maps
3.5 S1-Equivariant Localization

4 Basics of Comple X Morse Theory
4.1 Some Fundamental Constructions
4.2 Topological Applications of Lefschetz Pencils
4.3 The Hard Lefschetz Theorem
4.4 Vanishing Cycles and Local Monodromy
4.5 Proofofthe Picard Lefschetz formula
4.6 Global Picard-Lefschetz Formulae

5 Exercises and Solutions
5.1 Exercises
5.2 Solutions to Selected Exercises
References
Index

前言/序言

  As the the title suggests, the goal of this book is to give the reader a taste of the “unreasonable effectiveness” of Morse theory. The main idea behind thistechnique can be easily visualized.
  Suppose M is a smooth, compact manifold, which for simplicity we as-sume is embedded in a Euclidean space E. We would like to understand basictopological invariants of M such as its homology, and we attempt a “slicing” technique.
  We fix a unit vector u in E and we start slicing M with the family of hyperplanes perpendicular to u. Such a hyperplane will in general intersectM along a submanifold (slice). The manifold can be recovered by continuouslystacking the slices on top of each other in the same order as they were cut out of M.
  Think of the collection of slices as a deck of cards of various shapes. If welet these slices continuously pile up in the order they were produced, we noticean increasing stack of slices. As this stack grows, we observe that there aremoments of time when its shape suffers a qualitative change. Morse theoryis about extracting quantifiable information by studying the evolution of theshape of this growing stack of slices.

用戶評價

評分

莫爾斯理論

評分

微分拓撲學的一個重要分支。通常指兩部分內容:一是微分流形上可微函數的莫爾斯理論,即臨界點理論;二是變分問題的莫爾斯理論,即大範圍變分法。

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2.2 局部最短性

評分

封皮做工不咋,內容還是很好的

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2.2 局部最短性

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2.1 唯一性及存在性

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評分

好書

評分

不錯不錯不錯

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