Springer GTM 82.
"The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology. Accordingly, we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology."
Chapter I: de Rham TheoryEdit
Notes for Sections 1-6 (includes a deep treatment of vector bundles for section 5): https://themodularperspective.com/2019/03/25/graduate-text-notes-current-as-of-3-25-2019/
Section I.1: The de Rham Complex on $ \mathbb{R}^n $Edit
- Exercise 1.6 It wasn't immediately clear to me that
$ \displaystyle\frac{\Omega^1_c(\mathbb{R}^n)}{\ker \int_{\mathbb{R}}} \cong \mathbb{R}. $
To see this, assume that f dx and g dx are compactly-supported one-forms on R, and that
$ \int_{\mathbb{R}} f \, dx = \int_{\mathbb{R}} g \, dx. $
Then it follows that (f-g) dx is a compactly supported one-form (its support is contained in the union of supp f and supp g) with integral zero, and therefore
$ (f - g) \, dx \in \ker \int_{\mathbb{R}} = {\rm im}\, d. $
Therefore the equivalence class of a compactly-supported one-form in $ H_{c,DR}^1(\mathbb{R}) $ is completely determined by its integral.
Section I.2: The Mayer-Vietoris SequenceEdit
- The commutativity with d defines $ f^* $ uniquely: We need the assumption that $ f^* $ will be an $ \mathbb{R} $-algebra homomorphism here -- i.e., it should be $ \mathbb{R} $-linear and preserve multiplication.
http://math.stackexchange.com/questions/94691/why-is-the-pullback-completely-determined-by-d-f-ast-f-ast-d-in-de-rham-co/94702#94702