de Rham Cohomology
Definition. $H^k_{dR}(M) = \dfrac{\{\text{closed }k\text{-forms}\}}{\{\text{exact }k\text{-forms}\}}$ — closed forms modulo exact ones.
Intuition. Measures the global topology of $M$ using only calculus. On $\mathbb{R}^n$ every closed form is exact (Poincaré lemma) — all cohomology vanishes. On a space with holes, some closed forms fail to be exact, and each failure detects a hole: $H^1(\mathbb{R}^2 \setminus \{0\}) = \mathbb{R}$, generated by $d\theta = \frac{-y\,dx + x\,dy}{x^2+y^2}$ (closed, not exact — "the angle is only locally a function").
Physics anchor — twice. - $H^1$: in the field-free region outside a solenoid, $A$ is closed but not exact ⇒ Aharonov-Bohm Effect. - $H^2$: the class $\left[\frac{F}{2\pi}\right] \in H^2(M)$ of a $U(1)$ curvature is quantized (integer periods) and classifies the bundle ⇒ Dirac Monopole, Characteristic Classes.
Landau's formalism, being local, is structurally blind to both.
Related: Exterior Derivative · Stokes Theorem · Characteristic Classes · Aharonov-Bohm Effect · Module 02 - Differential Forms Refresher · Module 11 - Topology - Monopoles and Chern Classes
Sources: Sternberg Ch. III.