Logical dependence of the chapters of Lee's Smooth Manifolds, 2nd Ed.

Question

Has anyone written down the logical dependence graph for the chapters of Lee's wonderful book Smooth Manifolds, 2nd Ed.? I'm currently working my way through it and would like to know which chapters I could skip, if I so choose. My main goal is to understand the topology of Lie groups and de Rham cohomology, but of course I started by reading the first six chapters which cover basic facts about manifolds in general. I'm now part of the way through chapter 7 and plan to read chapters 8 and 9 next before perhaps skipping ahead.

Answer 1

For your purposes, you could get by with skipping Chapters 10 (Riemannian Metrics) and 22 (Symplectic Manifolds), but not much else.

The core of de Rham cohomology is covered in Chapters 17 and 18, which depend directly on the material in Chapters 14, 15, and 16, which in turn depend heavily on Chapters 9 through 12. If you only want to understand what de Rham cohomology is and how it is used in differential geometry, you could skip Chapter 18 (The de Rham Theorem), which connects it to algebraic topology.

Many of the deepest results about Lie groups and their applications are introduced and proved in Chapters 20 (The Exponential Map) and 21 (Quotient Manifolds), which depend directly on the material in Chapter 19 (Foliations). But if you're more interested in Lie groups as objects in their own right rather than the way they're used in differential geometry, you could stop after Chapter 20.