De Rham Cohomology

The exterior derivative satisfies $d^2=0$, so exact forms (those equal to $d\alpha$) are automatically closed ($d\omega =0$). The extent to which the converse fails — closed forms that are not exact — is a topological invariant of the manifold, the de Rham cohomology. This is where calculus becomes topology: it explains why $\oint \vec{A}\cdot d\vec{\ell }\ne 0$ around a solenoid, and it is the language of characteristic classes (chern-weil.md) and BRST cohomology. It builds on tensors-and-forms.md.

The de Rham complex

Because $d:{\Omega }^p(M)\to {\Omega }^{p+1}(M)$ squares to zero, the spaces of forms form a cochain complex $$0\to {\Omega }^0(M)\stackrel{d}{\to }{\Omega }^1(M)\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\Omega }^n(M)\to 0.$$ Define:

• closed forms $Z^p=\ker (d:{\Omega }^p\to {\Omega }^{p+1})$ — the cocycles;
• exact forms $B^p=\mathrm{im}(d:{\Omega }^{p-1}\to {\Omega }^p)$ — the coboundaries.

Since $d^2=0$, $B^p\subseteq Z^p$. The $p$-th de Rham cohomology is the quotient $$H_{\mathrm{dR}}^p(M)=\frac{Z^p}{B^p}=\frac{\text{closed }p\text{-forms}}{\text{exact }p\text{-forms}}.$$ Its dimension is the Betti number $b_p=\dim H_{\mathrm{dR}}^p(M)$. A nonzero class is a closed form that is not the derivative of anything — a global obstruction invisible to local calculus.

The Poincaré lemma

Locally there is no obstruction:

Poincaré lemma. On a contractible open set (e.g. a ball, or any star-shaped region), every closed form is exact: $H_{\mathrm{dR}}^p=0$ for $p\ge 1$.

So cohomology is entirely a global phenomenon — it measures how the manifold fails to be contractible. The proof constructs an explicit homotopy operator $K$ with $dK+Kd=\mathrm{id}$ on positive-degree forms.

Examples

• $H_{\mathrm{dR}}^0(M)\cong {\mathbb{R}}^{(\#\text{connected components})}$: closed $0$-forms are locally constant functions.
• The circle $S^1$: $H_{\mathrm{dR}}^1(S^1)\cong \mathbb{R}$, generated by the angle form $d\theta$ — closed but not exact ($\theta$ is not a global function). This is the winding number, and the mathematical content of the Aharonov–Bohm phase (see holonomy.md).
• Punctured plane ${\mathbb{R}}^2\setminus \{0\}$: $H^1\cong \mathbb{R}$, generated by $\frac{-ydx+xdy}{x^2+y^2}$ — the field of a magnetic vortex; $\oint =2\pi$ around the hole. This is why a solenoid produces a nonzero line integral through a field-free region.
• The $n$-sphere $S^n$: $H_{\mathrm{dR}}^p=\mathbb{R}$ for $p=0,n$ and $0$ otherwise — the top class is the volume form / area element.

The de Rham theorem

De Rham cohomology, defined by smooth analysis, computes a purely topological invariant:

De Rham theorem. For a smooth manifold, $H_{\mathrm{dR}}^p(M)\cong >H^p(M;\mathbb{R})$ — de Rham cohomology is isomorphic to the singular cohomology of the underlying topological space (real coefficients). The pairing is integration: $⟨[\omega ],[c]⟩={\int }_c\omega$ over cycles $c$.

Thus the same numbers $b_p$ arise whether one counts closed-mod-exact forms or $p$-dimensional "holes". Integration over cycles is well-defined on cohomology exactly because of Stokes' theorem (${\int }_{\partial c}\omega ={\int }_{c}d\omega$), pairing $d^2=0$ with ${\partial }^2=0$.

Why physics cares

• Characteristic classes (chern-weil.md) are de Rham classes built from curvature; their integrals over cycles are the quantized topological charges (Chern numbers, instanton number).
• Aharonov–Bohm / flux quantization is $H^1\ne 0$ of a non-contractible space (holonomy.md).
• BRST cohomology (gauge/brst.md) defines physical states as $\ker Q_B/\mathrm{im}{Q}_B$ — the same closed-mod-exact pattern with the BRST charge in place of $d$.

References

• Bott & Tu, Differential Forms in Algebraic Topology, Ch. 1 — the canonical reference.
• Lee, Introduction to Smooth Manifolds, Ch. 17.
• Nakahara, Geometry, Topology and Physics, Ch. 6.