The n-sphere

For the n-sphere, ${\displaystyle S^{n}}$, and also when taken together with a product of open intervals, we have the following. Let n > 0, m ≥ 0, and I be an open real interval. Then

${\displaystyle H_{\mathrm {dR} }^{k}(S^{n}\times I^{m})\simeq {\begin{cases}\mathbb {R} &k=0{\text{ or }}k=n,\\0&k\neq 0{\text{ and }}k\neq n.\end{cases}}}$

The n-torus

The ${\displaystyle n}$-torus is the Cartesian product: ${\displaystyle T^{n}=\underbrace {S^{1}\times \cdots \times S^{1}} _{n}}$. Similarly, allowing ${\displaystyle n\geq 1}$ here, we obtain

${\displaystyle H_{\mathrm {dR} }^{k}(T^{n})\simeq \mathbb {R} ^{n \choose k}.}$

We can also find explicit generators for the de Rham cohomology of the torus directly using differential forms. Given a quotient manifold ${\displaystyle \pi :X\to X/G}$ and a differential form ${\displaystyle \omega \in \Omega ^{k}(X)}$ we can say that ${\displaystyle \omega }$ is ${\displaystyle G}$-invariant if given any diffeomorphism induced by ${\displaystyle G}$, ${\displaystyle \cdot g:X\to X}$ we have ${\displaystyle (\cdot g)^{*}(\omega )=\omega }$. In particular, the pullback of any form on ${\displaystyle X/G}$ is ${\displaystyle G}$-invariant. Also, the pullback is an injective morphism. In our case of ${\displaystyle \mathbb {R} ^{n}/\mathbb {Z} ^{n}}$ the differential forms ${\displaystyle dx_{i}}$ are ${\displaystyle \mathbb {Z} ^{n}}$-invariant since ${\displaystyle d(x_{i}+k)=dx_{i}}$. But, notice that ${\displaystyle x_{i}+\alpha }$ for ${\displaystyle \alpha \in \mathbb {R} }$ is not an invariant ${\displaystyle 0}$-form. This with injectivity implies that

${\displaystyle [dx_{i}]\in H_{dR}^{1}(T^{n})}$

Since the cohomology ring of a torus is generated by ${\displaystyle H^{1}}$, taking the exterior products of these forms gives all of the explicit representatives for the de Rham cohomology of a torus.

Punctured Euclidean space

Punctured Euclidean space is simply ${\displaystyle \mathbb {R} ^{n}}$ with the origin removed.

${\displaystyle H_{\text{dR}}^{k}(\mathbb {R} ^{n}\setminus \{0\})\cong {\begin{cases}\mathbb {R} ^{2}&n=1,k=0\\\mathbb {R} &n>1,k=0,n-1\\0&{\text{otherwise}}\end{cases}}.}$

The Möbius strip

We may deduce from the fact that the Möbius strip, M, can be deformation retracted to the 1-sphere (i.e. the real unit circle), that:

${\displaystyle H_{\mathrm {dR} }^{k}(M)\simeq H_{\mathrm {dR} }^{k}(S^{1}).}$

Proof

The standard proof proceeds by showing that the de Rham complex, when viewed as a complex of sheaves, is an acyclic resolution of ${\textstyle {\underline {\mathbb {R} }}}$. In more detail, let m be the dimension of M and let ${\textstyle \Omega ^{k}}$ denote the sheaf of germs of ${\displaystyle k}$-forms on M (with ${\textstyle \Omega ^{0}}$ the sheaf of ${\textstyle C^{\infty }}$ functions on M). By the Poincaré lemma, the following sequence of sheaves is exact (in the abelian category of sheaves):

${\displaystyle 0\to {\underline {\mathbb {R} }}\to \Omega ^{0}\,\xrightarrow {d_{0}} \,\Omega ^{1}\,\xrightarrow {d_{1}} \,\Omega ^{2}\,\xrightarrow {d_{2}} \dots \xrightarrow {d_{m-1}} \,\Omega ^{m}\to 0.}$

This long exact sequence now breaks up into short exact sequences of sheaves

${\displaystyle 0\to \mathrm {im} \,d_{k-1}\,\xrightarrow {\subset } \,\Omega ^{k}\,\xrightarrow {d_{k}} \,\mathrm {im} \,d_{k}\to 0,}$

where by exactness we have isomorphisms ${\textstyle \mathrm {im} \,d_{k-1}\cong \mathrm {ker} \,d_{k}}$ for all k. Each of these induces a long exact sequence in cohomology. Since the sheaf ${\textstyle \Omega ^{0}}$ of ${\textstyle C^{\infty }}$ functions on M admits partitions of unity, any ${\textstyle \Omega ^{0}}$-module is a fine sheaf; in particular, the sheaves ${\textstyle \Omega ^{k}}$ are all fine. Therefore, the sheaf cohomology groups ${\textstyle H^{i}(M,\Omega ^{k})}$ vanish for ${\textstyle i>0}$ since all fine sheaves on paracompact spaces are acyclic. So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. At one end of the chain is the sheaf cohomology of ${\textstyle {\underline {\mathbb {R} }}}$ and at the other lies the de Rham cohomology.

Harmonic forms

If M is a compact Riemannian manifold, then each equivalence class in ${\displaystyle H_{\mathrm {dR} }^{k}(M)}$ contains exactly one harmonic form. That is, every member ${\displaystyle \omega }$ of a given equivalence class of closed forms can be written as

${\displaystyle \omega =\alpha +\gamma }$

where ${\displaystyle \alpha }$ is exact and ${\displaystyle \gamma }$ is harmonic: ${\displaystyle \Delta \gamma =0}$.

Any harmonic function on a compact connected Riemannian manifold is a constant. Thus, this particular representative element can be understood to be an extremum (a minimum) of all cohomologously equivalent forms on the manifold. For example, on a 2-torus, one may envision a constant 1-form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length). In this case, there are two cohomologically distinct combings; all of the others are linear combinations. In particular, this implies that the 1st Betti number of a 2-torus is two. More generally, on an ${\displaystyle n}$-dimensional torus ${\displaystyle T^{n}}$, one can consider the various combings of ${\displaystyle k}$-forms on the torus. There are ${\displaystyle n}$ choose ${\displaystyle k}$ such combings that can be used to form the basis vectors for ${\displaystyle H_{\text{dR}}^{k}(T^{n})}$; the ${\displaystyle k}$-th Betti number for the de Rham cohomology group for the ${\displaystyle n}$-torus is thus ${\displaystyle n}$ choose ${\displaystyle k}$.

More precisely, for a differential manifold M, one may equip it with some auxiliary Riemannian metric. Then the Laplacian ${\displaystyle \Delta }$ is defined by

${\displaystyle \Delta =d\delta +\delta d}$

with ${\displaystyle d}$ the exterior derivative and ${\displaystyle \delta }$ the codifferential. The Laplacian is a homogeneous (in grading) linear differential operator acting upon the exterior algebra of differential forms: we can look at its action on each component of degree ${\displaystyle k}$ separately.

If ${\displaystyle M}$ is compact and oriented, the dimension of the kernel of the Laplacian acting upon the space of k-forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree ${\displaystyle k}$: the Laplacian picks out a unique harmonic form in each cohomology class of closed forms. In particular, the space of all harmonic ${\displaystyle k}$-forms on ${\displaystyle M}$ is isomorphic to ${\displaystyle H^{k}(M;\mathbb {R} ).}$ The dimension of each such space is finite, and is given by the ${\displaystyle k}$-th Betti number.

Hodge decomposition

Let ${\displaystyle M}$ be a compact oriented Riemannian manifold. The Hodge decomposition states that any ${\displaystyle k}$-form on ${\displaystyle M}$ uniquely splits into the sum of three L² components:

${\displaystyle \omega =\alpha +\beta +\gamma ,}$

where ${\displaystyle \alpha }$ is exact, ${\displaystyle \beta }$ is co-exact, and ${\displaystyle \gamma }$ is harmonic.

One says that a form ${\displaystyle \beta }$ is co-closed if ${\displaystyle \delta \beta =0}$ and co-exact if ${\displaystyle \beta =\delta \eta }$ for some form ${\displaystyle \eta }$, and that ${\displaystyle \gamma }$ is harmonic if the Laplacian is zero, ${\displaystyle \Delta \gamma =0}$. This follows by noting that exact and co-exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms. Here, orthogonality is defined with respect to the L² inner product on ${\displaystyle \Omega ^{k}(M)}$:

${\displaystyle (\alpha ,\beta )=\int _{M}\alpha \wedge {\star \beta }.}$

By use of Sobolev spaces or distributions, the decomposition can be extended for example to a complete (oriented or not) Riemannian manifold.^[6]