目錄
1 Basic Topology
1.1 Metric Spaces
1.1.1 Metric Spaces
1.1.2 Multiplinear Algebra
1.1.3 Differential Forms
1.1.4 The de Rham Cohomology Groups on Rn
1.1.5 Stokes' Theorem in Rn
1.1.6 Brouwer's Fixed Point Theorem
1.1.7 Jordan Curve Theorem
1.1.8 Whitney-Graustein Theorem
1.2 Topological Spaces
1.2.1 Continuous Maps and Bases
1.2.2 Categories: An Introduction
1.2.3 Subspaces
1.2.4 Separation Axioms
1.2.5 Topological Manifolds
1.2.6 Brouwer's Theorem on the Invariance of Topological Dimensions
1.2.7 Cantor Sets and Alexandroff's Theorem
1.3 Constructions
1.3.1 Product Spaces
1.3.2 Quotient Spaces
1.4 Topological Groups and Matrix Lie Groups
1.4.1 Subgroups and Left/Right Translations
1.4.2 Group Actions
1.4.3 Matrix Lie Groups
1.5 Connectedness and Compactness
1.5.1 Connectedness
1.5.2 Path-Connectedness
1.5.3 Compactness
1.5.4 Paracompact Spaces
2 Analysis on Topological Groups
2.1 Haar Measures
2.1.1 Examples of Haar Measures
2.1.2 Convolution
2.1.3 Modular Functions on G
2.2 Iwasawa's Decomposition
2.2.1 Iwasawa's Decomposition
2.2.2 Characters
2.2.3 K-Bi-Invariant Functions
2.2.4 Group Fubini Theorem
2.3 Harish, Mellin and Spherical Transforms
2.3.1 The Harish Transform and the Orbital Integral
2.3.2 The Mellin and Spherical Transforms
2.3.3 Computation of the Orbital Integral
2.3.4 Gaussians on G and Their Spherical Transform
2.3.5 The Polar Haar Measure and Inversion
2.3.6 Point-Pair Invariants, the Polar Height, and the Polar Distance
3 Fundamental Groups
3.1 Homotopy
3.1.1 Homotopy Relation
3.1.2 The Fundamental Group
3.1.3 Fundamental Groups of Manifolds
3.1.4 Fundamental Groups of Product Spaces
3.1.5 Retractions and Deformation Retractions
3.1.6 Contractible Spaces
3.2 H-Spaces
3.2.1 Two Notions on Functors
3.2.2 H-Spaces
3.2.3 A Classical Example of H-Groups
3.2.4 Suspension
3.2.5 A Classical Example of H-Cogroups
3.2.6 Higher Homotopy Groups
3.3 π1(S1)
3.3.1 The Fundamental Groupoid
3.3.2 The Equivalent of π1
3.3.3 π1(S1)=Z
Main References
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