Topology

  1. Differential Topology
    1. Smooth Manifolds
      1. Topological Manifolds
        1. Definition and Examples
          1. Hausdorff and Second Countable Conditions
          2. Smooth Structures
            1. Charts and Coordinate Systems
              1. Atlases and Compatibility
                1. Smooth Atlases
                  1. Maximal Atlases
                  2. Examples of Smooth Manifolds
                    1. Euclidean Space
                      1. Spheres
                        1. Tori
                          1. Projective Spaces
                            1. Grassmannians
                              1. Lie Groups
                              2. Smooth Maps
                                1. Definition between Manifolds
                                  1. Local Coordinate Representations
                                    1. Diffeomorphisms
                                  2. Tangent Spaces and Derivatives
                                    1. Tangent Vectors
                                      1. Geometric Approach
                                        1. Algebraic Approach via Derivations
                                          1. Equivalence of Definitions
                                          2. Tangent Spaces
                                            1. Definition and Properties
                                              1. Dimension of Tangent Space
                                              2. Differential of Smooth Maps
                                                1. Definition and Properties
                                                  1. Chain Rule
                                                    1. Local Coordinate Formula
                                                    2. Tangent Bundle
                                                      1. Definition as Disjoint Union
                                                        1. Smooth Structure on Tangent Bundle
                                                          1. Vector Bundle Structure
                                                        2. Submanifolds
                                                          1. Regular Level Sets
                                                            1. Regular Values
                                                              1. Preimage Theorem
                                                              2. Embedded Submanifolds
                                                                1. Definition and Examples
                                                                  1. Whitney Embedding Theorem
                                                                  2. Immersed Submanifolds
                                                                    1. Immersions and Embeddings
                                                                      1. Examples of Immersions
                                                                      2. Sard's Theorem
                                                                        1. Statement and Applications
                                                                          1. Generic Properties
                                                                        2. Vector Fields and Flows
                                                                          1. Vector Fields
                                                                            1. Definition and Examples
                                                                              1. Smooth Vector Fields
                                                                                1. Vector Fields as Sections
                                                                                2. Integral Curves
                                                                                  1. Existence and Uniqueness
                                                                                    1. Maximal Integral Curves
                                                                                    2. Flows
                                                                                      1. One-Parameter Groups
                                                                                        1. Complete Vector Fields
                                                                                          1. Lie Derivatives
                                                                                        2. Differential Forms
                                                                                          1. Cotangent Spaces
                                                                                            1. Linear Functionals on Tangent Spaces
                                                                                              1. Cotangent Bundle
                                                                                              2. Differential 1-Forms
                                                                                                1. Definition and Examples
                                                                                                  1. Pullback of Forms
                                                                                                  2. Higher Degree Forms
                                                                                                    1. Alternating Multilinear Forms
                                                                                                      1. Wedge Product
                                                                                                        1. Exterior Algebra
                                                                                                        2. Exterior Derivative
                                                                                                          1. Definition and Properties
                                                                                                            1. de Rham Complex
                                                                                                              1. Closed and Exact Forms
                                                                                                              2. Integration of Forms
                                                                                                                1. Orientation of Manifolds
                                                                                                                  1. Integration on Oriented Manifolds
                                                                                                                    1. Stokes' Theorem
                                                                                                                    2. de Rham Cohomology
                                                                                                                      1. Definition and Basic Properties
                                                                                                                        1. Relationship to Singular Cohomology
                                                                                                                      2. Morse Theory
                                                                                                                        1. Morse Functions
                                                                                                                          1. Critical Points
                                                                                                                            1. Non-Degenerate Critical Points
                                                                                                                              1. Morse Lemma
                                                                                                                              2. Index of Critical Points
                                                                                                                                1. Hessian Matrix
                                                                                                                                  1. Index and Nullity
                                                                                                                                  2. Morse Inequalities
                                                                                                                                    1. Relationship to Homology
                                                                                                                                      1. Applications to Topology
                                                                                                                                      2. Handle Decompositions
                                                                                                                                        1. Attaching Handles
                                                                                                                                          1. CW Structure from Morse Functions

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                                                                                                                                      5. Geometric Topology