An introduction to manifolds by Loring Tu in 2010
1. Euclidean spaces
1. Smooth functions on a Euclidean space
2. Tangent vectors in \[\mathbb{R}^n\] as derivations
3. The exterior algebra of multicovectors
2. Manifolds
5. Manifolds
6. Smooth maps on a manifold
7. Quotients
3. The tangent space
8. The tangent space
9. Submanifolds
10. Categories and functors
11. The rank of a smooth map
12. The tangent bundle
13. Bump functions and partitions of unity
14. Vector fields
4. Lie groups and Lie algebras
15. Lie groups
16. Lie algebras
19. The exterior derivative
20. The Lie derivative and interior multiplication
6. Integration
21. Orientations
22. Manifolds with boundary
23. Integration on Manifolds
7. De Rham Theory
24. De Rham Cohomology
25. The Long exact sequence in cohomology
26. The Mayer-Vietoris sequence
27. Homotopy invariance
28. Computation of de Rham Cohomology
29. Proof of homotopy invariance
Appendices
A. Point-Set Topology
C. Existence of a partition of unity in general
D. Linear algebra
E. Quaternions and the symplectic group