I have followed Spivak rather ¡­

I have followed Spivak rather closely until page 265, but after this I do things differently. After giving a somewhat different proof of Corollary 8, p.266, I proved the homotopy invariance of de Rham cohomology (Theorem 13, p.277) and jumped to chapter 11 and construction of the Mayer-Vietoris sequence. As an illustration I calculated the cohomology of spheres and presented two applications: Brouwer's fixed point theorem and the proof that a sphere admits a nonsingular vector field if and only if the dimension is odd. In the remaining time I will continue to study de Rham cohomology. Most of the results will be found in Spivak's book, but the treatment will differ in some places. Supplementary notes will be found here.