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De Rham cohomology
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De Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship.
The de Rham complex is the cochain complex of differential forms on some smooth manifold M, with the exterior derivative as the differential:
where Ω0(M) is the space of smooth functions on M, Ω1(M) is the space of 1-forms, and so forth. Forms that are the image of other forms under the exterior derivative, plus the constant 0 function in Ω0(M), are called exact and forms whose exterior derivative is 0 are called closed (see Closed and exact differential forms); the relationship d2 = 0 then says that exact forms are closed.
In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and the 1-form corresponding to the derivative of angle from a reference point at its centre, typically written as dθ (described at Closed and exact differential forms). There is no function θ defined on the whole circle such that dθ is its derivative; the increase of 2π in going once around the circle in the positive direction implies a multivalued function θ. Removing one point of the circle obviates this, at the same time changing the topology of the manifold.
One prominent example when all closed forms are exact is when the underlying space is contractible to a point or, more generally, if it is simply connected (no-holes condition). In this case the exterior derivative restricted to closed forms has a local inverse called a homotopy operator. Since it is also nilpotent, it forms a dual chain complex with the arrows reversed compared to the de Rham complex. This is the situation described in the Poincaré lemma.
The idea behind de Rham cohomology is to define equivalence classes of closed forms on a manifold. One classifies two closed forms α, β ∈ Ωk(M) as cohomologous if they differ by an exact form, that is, if α − β is exact. This classification induces an equivalence relation on the space of closed forms in Ωk(M). One then defines the k-th de Rham cohomology group to be the set of equivalence classes, that is, the set of closed forms in Ωk(M) modulo the exact forms.
Note that, for any manifold M composed of m disconnected components, each of which is connected, we have that
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De Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship.
The de Rham complex is the cochain complex of differential forms on some smooth manifold M, with the exterior derivative as the differential:
where Ω0(M) is the space of smooth functions on M, Ω1(M) is the space of 1-forms, and so forth. Forms that are the image of other forms under the exterior derivative, plus the constant 0 function in Ω0(M), are called exact and forms whose exterior derivative is 0 are called closed (see Closed and exact differential forms); the relationship d2 = 0 then says that exact forms are closed.
In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and the 1-form corresponding to the derivative of angle from a reference point at its centre, typically written as dθ (described at Closed and exact differential forms). There is no function θ defined on the whole circle such that dθ is its derivative; the increase of 2π in going once around the circle in the positive direction implies a multivalued function θ. Removing one point of the circle obviates this, at the same time changing the topology of the manifold.
One prominent example when all closed forms are exact is when the underlying space is contractible to a point or, more generally, if it is simply connected (no-holes condition). In this case the exterior derivative restricted to closed forms has a local inverse called a homotopy operator. Since it is also nilpotent, it forms a dual chain complex with the arrows reversed compared to the de Rham complex. This is the situation described in the Poincaré lemma.
The idea behind de Rham cohomology is to define equivalence classes of closed forms on a manifold. One classifies two closed forms α, β ∈ Ωk(M) as cohomologous if they differ by an exact form, that is, if α − β is exact. This classification induces an equivalence relation on the space of closed forms in Ωk(M). One then defines the k-th de Rham cohomology group to be the set of equivalence classes, that is, the set of closed forms in Ωk(M) modulo the exact forms.
Note that, for any manifold M composed of m disconnected components, each of which is connected, we have that
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