Intereting Posts

Evans PDE Chapter 5 Problem 11: Does $Du=0$ a.e. implie $u=c$ a.e.?
Proving $\ell^p$ is complete
Is symmetric group on natural numbers countable?
$Ax=b\Leftrightarrow b\in\left(\ker A^*\right)^\perp$
Difference between proof and plausible argument.
Every finitely generated algebra over a field is a Jacobson ring
Describe $R=\mathbb{Z}/(X^2-3,2X+4)$
$G$ is a nonabelian finite group, then $|Z(G)|\leq \frac{1}{4}|G|$
Problem about jointly continuous and linearity of expectation.
Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.
Minimization of $\sum \frac{1}{n_k}\ln n_k >1 $ subject to $\sum \frac{1}{n_k}\simeq 1$
A ‘strong’ form of the Fundamental Theorem of Algebra
How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?
Calculate the p.m.f. of a non-monotonous function of a random variable
Prove that a bounded sequence has two convergent subsequences.

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between the top de Rham cohomology and the integral $\int_M \omega$ of the members of each equivalence class $\omega$.

Moving on to de Rham cohomology below the top one. Taking the example of the torus as in http://topospaces.subwiki.org/wiki/Torus , is it possible to link the rank $\binom{n}{k}$ to the values of specific integrals in a similar fashion ?

- Is line element mathematically rigorous?
- Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
- How to calculate gradient of $x^TAx$
- Is there an intuitve motivation for the wedge product in differential geometry?
- What is the intuition behind the definition of the differential of a function?
- Exercise concerning the Lefschetz fixed point number

- $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$
- Boundary of product manifolds such as $S^2 \times \mathbb R$
- A smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points.
- Diameter of the Grassmannian
- Integration of a differential form along a curve
- How do I know when a form represents an integral cohomology class?
- Does the Riemann tensor encode all information about the second derivatives of the metric?
- Elementary proof of the fact that any orientable 3-manifold is parallelizable
- Trouble understanding proof of this proposition on contact type hypersurfaces
- Which manifolds are parallelizable?

Let $T^{n}$ denote the $n$ torus* and $\pi_{i} : T^{n} \rightarrow S^{1}$ denote the $i-th$ projection. Let us denote the pull-back of the top from (if you do not take the normalized top form i.e. $\int_{s^{1}} d\theta = 1$, you will get some $2\pi$ factors) $\pi_{i}^{\ast}(d\theta) = \omega_{i}$. Then each $\omega_{i}\in H^{1}_{DR}(T^{n})$ and by dimension count and the fact that they are linearly independent (as can be seen by integrating on a ‘suitable’ component $S^{1}$) they together generate $H^{1}_{DR}(T^{n})$.

Now assume the ranks of all de-Rham cohomology of $T^{n}$ are known. The cohomologies $H^{k}(T^{n})$ are generated by wedge of $k-$ subsets of $\omega_{i}$. That they are linearly independent (in cohomology) can be seen by integrating over $k-$ cycles consisting respective component $S^{1}$ factors. Then dimension count gives isomorphism. (There is a more correct proof by showing compatability of wedge and cup products and using comparison of singular/simplicial cohomology and de-Rahm cohomology).

The inegral of each such wedge is just a product of constituent $\omega_{i}$ on $(S^{1})_{i}$.

Hence the basis element $\sum_{I \subset \{1,2,..n\}, |I| = k} (\wedge_{i \in I}\omega_{i})$ integrates on the k-cycle $\sum_{I \subset \{1,2,..n\}, |I| = k}\pm [\prod_{i \in I}(S^{1})_{i}]$ to give you the ranks. The plus minuses are important due to rule of signs in wedge products.

*Torus as per your question. Too many things are called torus in too many different contexts.

- Proove that $2005$ devides $55555\dots$ with 800 5's
- How is $\mathbb R^2\setminus \mathbb Q^2$ path connected?
- Closure of a connected set is connected
- What is the relation between $ \kappa$-monolithic and monotonically monolithic?
- why is the following thing a projection operator?
- Philosophy or meaning of adjoint functors
- weak subsolution
- When is a cyclotomic polynomial over a finite field a minimal polynomial?
- Localization of the Integer Ring
- Product of two power series
- On the number of ways of writing an integer as a sum of 3 squares using triangular numbers.
- mid-point convex but not a.e. equal to a convex function
- How can I find all solutions to differential equation?
- How do I show that $\int_{-\infty}^\infty \frac{ \sin x \sin nx}{x^2} \ dx = \pi$?
- What would happen if ZFC were found to be inconsistent?