9.4 De Rham cohomology

De Rham cohomology connects differential forms to the topology of smooth manifolds. It uses the exterior derivative to build a complex of forms, measuring how closed forms differ from exact ones.

This approach reveals geometric and topological properties of manifolds through analytic tools. It bridges the gap between calculus on manifolds and algebraic topology, providing insights into the structure of spaces.

Differential Forms and Exterior Derivative

Constructing Differential Forms

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• Differential forms generalize the concept of functions and differential operators to arbitrary dimensions
• Construct differential $k$-forms on a smooth manifold $M$ by taking the wedge product of $k$ differential 1-forms ($dx^i$)
• Differential forms provide a way to integrate over submanifolds of $M$ (curves, surfaces, etc.)
• Allow for a coordinate-free description of geometric structures on manifolds (volume forms, symplectic forms)

Properties of the Exterior Derivative

• Exterior derivative $d$ maps $k$-forms to $(k+1)$-forms, generalizing the concept of the differential of a function
• Satisfies the property $d^2 = 0$, which is analogous to the equality of mixed partial derivatives
• Closed forms defined as differential forms $\omega$ satisfying $d\omega = 0$ ($\omega$ has vanishing exterior derivative)
• Exact forms defined as differential forms $\omega$ that can be written as $\omega = d\eta$ for some $(k-1)$-form $\eta$
• Exact forms are always closed due to the property $d^2 = 0$ (closed forms may not be exact)

De Rham Cohomology

De Rham Complex and Cohomology Groups

• De Rham complex is the sequence of vector spaces of differential forms connected by the exterior derivative $d$
• Cohomology groups $H^k_{dR}(M)$ measure the failure of the Poincaré lemma globally on the manifold $M$
• $H^k_{dR}(M)$ defined as the quotient of the space of closed $k$-forms by the space of exact $k$-forms ($\ker d / \operatorname{im} d$)
• Intuitively, cohomology groups capture the "holes" in the manifold that prevent closed forms from being exact

Poincaré Lemma and De Rham Theorem

• Poincaré lemma states that on a contractible open subset $U$ of $M$, every closed form is exact ($H^k_{dR}(U) = 0$ for $k > 0$)
• Provides a local characterization of the cohomology groups (they vanish on contractible sets)
• De Rham theorem establishes an isomorphism between the de Rham cohomology groups and the singular cohomology groups with real coefficients
• Connects the purely analytic theory of differential forms with the topological invariants of the manifold (Betti numbers)

Cohomological Tools

Mayer-Vietoris Sequence

• Mayer-Vietoris sequence relates the cohomology of a manifold $M$ to the cohomology of two open subsets $U$ and $V$ covering $M$
• Provides a tool for computing cohomology groups by decomposing the manifold into simpler pieces (e.g., contractible sets)
• Sequence involves the cohomology groups of $U$, $V$, $U \cap V$, and $M$, connected by restriction and difference maps
• Useful for proving the Poincaré lemma by induction on the dimension of the manifold

Künneth Formula and Hodge Theory

• Künneth formula relates the cohomology of a product manifold $M \times N$ to the cohomology of the factors $M$ and $N$
• Allows for the computation of cohomology groups of higher-dimensional manifolds from lower-dimensional ones (e.g., tori from circles)
• Hodge theory studies the relationship between de Rham cohomology and the Laplacian operator on differential forms
• Establishes a decomposition of differential forms into harmonic, exact, and co-exact components (Hodge decomposition)
• Harmonic forms represent the cohomology classes and satisfy both $d\omega = 0$ and $d^*\omega = 0$ ($d^*$ is the adjoint of $d$)