de Rham cohomology

Let $X$ be a paracompact ${𝒞}^{\mathrm{\infty }}$ differential manifold. Let

$$\mathrm{\Omega }X=\underset{i=0}{\overset{\mathrm{\infty }}{\oplus }}{\mathrm{\Omega }}^{i}X$$

denote the graded-commutative $ℝ$-algebra of differential forms on $X$. Together with the exterior derivative

$$d^i:{\mathrm{\Omega }}^{i}X\to {\mathrm{\Omega }}^{i+1}X\mathit{\hspace{1em}}(i=0,1,\mathrm{\dots }),$$

$\mathrm{\Omega }X$ forms a chain complex $(\mathrm{\Omega }X,d)$ of $ℝ$-vector spaces. The ${\mathrm{H}}_{\mathrm{dR}}^{i}X$ of $X$ are defined as the homology groups of this complex, that is to say

$${\mathrm{H}}_{\mathrm{dR}}^{i}X:=(\ker d^i)/(im{d}^{i-1})\mathit{\hspace{1em}}(i=0,1,\mathrm{\dots }),$$

where ${\mathrm{\Omega }}^{-1}X$ is taken to be 0, so $d^{-1}:0\to {\mathrm{\Omega }}^{0}X$ is the zero map. The wedge product in $\mathrm{\Omega }X$ induces the structure of a graded-commutative $ℝ$-algebra on

$${\mathrm{H}}_{\mathrm{dR}}X:=\underset{i=0}{\overset{\mathrm{\infty }}{\oplus }}{\mathrm{H}}_{\mathrm{dR}}^{i}X.$$

If $X$ and $Y$ are both paracompact ${𝒞}^{\mathrm{\infty }}$ manifolds and $f:X\to Y$ is a differentiable map, there is an induced map

$$f^{*}:{\mathrm{H}}_{\mathrm{dR}}Y\to {\mathrm{H}}_{\mathrm{dR}}X,$$

defined by

$$f^{*}[\omega ]:=[f^{*}\omega ]\mathit{\hspace{1em}}\text{for }\omega \in \ker d.$$

Here $[\omega ]$ denotes the class of $\omega$ modulo $imd$, and the second $f^{*}$ is the map $\mathrm{\Omega }Y\to \mathrm{\Omega }X$ induced by the functor $\mathrm{\Omega }$. This action on differentiable maps makes the de Rham cohomology into a contravariant functor from the category of paracompact ${𝒞}^{\mathrm{\infty }}$ manifolds to the category of graded-commutative $ℝ$-algebras. It turns out to be homotopy invariant; this implies that homotopy equivalent manifolds have isomorphic de Rham cohomology.