De Rham cohomology measures the global failure of closed forms to be exact. It converts differential equations into topological invariants while retaining concrete representatives that can be integrated.
The de Rham complex
Definition. Since \(\mathrm{d}^2=0\), the sequence \[ 0\to\Omega^0(M)\xrightarrow{\mathrm{d}}\Omega^1(M) \xrightarrow{\mathrm{d}}\cdots\xrightarrow{\mathrm{d}}\Omega^n(M)\to0 \] is a cochain complex. Its \(k\)-th cohomology is \[ H_{\mathrm{dR}}^k(M) =\frac{\ker(\mathrm{d}:\Omega^k\to\Omega^{k+1})} {\operatorname{im}(\mathrm{d}:\Omega^{k-1}\to\Omega^k)}. \]
The wedge product descends to cohomology because \(\mathrm{d}\) is a graded derivation, making \(H_{\mathrm{dR}}^\bullet(M)\) a graded-commutative algebra.
Proposition. If \(M\) is connected, \(H_{\mathrm{dR}}^0(M)\cong\mathbb R\).
Proof. A zero-form \(f\) is closed exactly when \(\mathrm{d}f=0\), hence when it is locally constant. On a connected manifold it is constant. There are no exact zero-forms other than zero.
Homotopy invariance
Theorem. Smoothly homotopic maps \(f_0,f_1:M\to N\) induce the same homomorphism on de Rham cohomology. Consequently homotopy-equivalent manifolds have isomorphic de Rham cohomology.
Proof. For a homotopy \(F:M\times[0,1]\to N\), integration along the interval defines an operator \(K:\Omega^k(N)\to\Omega^{k-1}(M)\) satisfying \[ f_1^*-f_0^*=\mathrm{d}K+K\mathrm{d}. \] If \(\omega\) is closed, the difference \(f_1^*\omega-f_0^*\omega=\mathrm{d}(K\omega)\) is exact, so the induced cohomology classes agree.
Computations
Example (circle). Every one-form on \(S^1\) is closed. The map \[ [\omega]\longmapsto\int_{S^1}\omega \] is an isomorphism \(H_{\mathrm{dR}}^1(S^1)\cong\mathbb R\). Its kernel consists of exact forms: lift \(\omega\) to a periodic form \(a(t)\,\mathrm{d}t\) and integrate \(a\); zero period makes the primitive periodic.
Theorem (Mayer-Vietoris). If \(M=U\cup V\), there is a long exact sequence \[ \cdots\to H^{k-1}(U\cap V)\to H^k(M)\to H^k(U)\oplus H^k(V)\to H^k(U\cap V)\to\cdots. \]
A partition of unity produces the cochain-level splitting behind this sequence. Applying it to two contractible arcs of \(S^1\), or to hemispheres of \(S^n\), gives \[ H_{\mathrm{dR}}^k(S^n)\cong \begin{cases} \mathbb R,&k=0,n,\\ 0,&\text{otherwise}. \end{cases} \]
Integration and topology
Theorem (de Rham). Integration over smooth singular cycles induces an isomorphism \[ H_{\mathrm{dR}}^k(M)\cong H^k_{\mathrm{sing}}(M;\mathbb R). \]
Thus a differential invariant defined using charts and derivatives depends only on the underlying topology. For a compact oriented \(n\)-manifold, integration identifies \(H_{\mathrm{dR}}^n(M)\cong\mathbb R\), and Poincar? duality pairs \(H^k\) nondegenerately with \(H^{n-k}\) through \(([\alpha],[\beta])\mapsto\int_M\alpha\wedge\beta\).