# Lecture Notes

LEC # TOPICS  LECTURE NOTES
1 Manifolds: Definitions and Examples (PDF)
2 Smooth Maps and the  Notion of Equivalence

Standard Pathologies

(PDF)
3 The Derivative of a Map between Vector Spaces (PDF)
4 Inverse and Implicit Function Theorems (PDF)
5 More Examples (PDF)
6 Vector Bundles and the Differential: New Vector Bundles from Old (PDF)
7 Vector Bundles and the Differential: The Tangent Bundle (PDF)
8 Connections

Partitions of Unity

The Grassmanian is Universal

(PDF)
9 The Embedding Manifolds in RN (PDF)
10-11 Sard’s Theorem (PDF)
12 Stratified Spaces (PDF)
13 Fiber Bundles (PDF)
14 Whitney’s Embedding Theorem, Medium Version (PDF)
15 A Brief Introduction to Linear Analysis: Basic Definitions

A Brief Introduction to Linear Analysis: Compact Operators

(PDF)
16-17 A Brief Introduction to Linear Analysis: Fredholm Operators (PDF)
18-19 Smale’s Sard Theorem (PDF)
20 Parametric Transversality (PDF)
21-22 The Strong Whitney Embedding Theorem (PDF)
23-28 Morse Theory (PDF)
29 Canonical Forms: The Lie Derivative (PDF)
30 Canonical Forms: The Frobenious Integrability Theorem

Canonical Forms: Foliations

Characterizing a Codimension One Foliation in Terms of its Normal Vector

The Holonomy of Closed Loop in a Leaf

Reeb’s Stability Theorem

(PDF)
31 Differential Forms and de Rham’s Theorem: The Exterior Algebra (PDF)
32 Differential Forms and de Rham’s Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology

Cech Cohomology

(PDF)
33 Refinement The Acyclicity of the Sheaf of p-forms (PDF)
34 The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology (PDF)
35 The Immersion Theorem of Smale (PDF)

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