# Questions tagged [de-rham-cohomology]

This a cohomogy theory for smooth manifolds, where the chain complex is defined by "closed n-forms" / "exact n-forms".

121 questions
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### Cup and wedge product in singular and de Rham cohomology

De Rham's theorem asserts that the map $I: H_{dR}^p(M) \to H_{sing}^p(M, \mathbb{R})$ defined as $$I(\omega)= [\sigma^p] \mapsto\int_{\sigma^p}\omega$$ is an isomorphism ($\sigma^p \in [\sigma^p]$ ...
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### First De Rham cohomology group of Cantor ternary

I proved that $\mathcal H^1 (\mathbb R^2)=\mathbb 0$ and if $\{ P_1,...,P_n\ |P_i\neq P_j,\ i\neq j \}$ is an $\mathbb R^2$ subset, $\mathcal H^1 (\mathbb R^2\setminus\{P_1,...,P_n\})=\mathbb R^n$. ...
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### Cauchy's Theorem, Stokes' Theorem, de Rham Cohomology

I've been struggling these last couple of days to see the connection, if at all there is one, between the following facts: For holomorphic functions $f$, $\mathrm{d}(f(z)\mathrm{d}z) = 0$. In a ...
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### de Rham isomorphism

Let $X$ be an open manifold, with one end $N$, Q How to show that $H^1_{c,dR}(X)\to H^1_{dR}(X)$ is an injective,? here $H^1_{dR}$ denotes the de Rham cohomology and $H^1_{c,dR}$ denotes the ...
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### Show that $\mathbb{S}^{n+m}$ is not homeomorphic to a product of orientable manifolds

I want to prove that the sphere $\mathbb{S}^{n+m}$ is not homeomorphic to the product of N and M, orientable manifolfs with $\textit{dim}\;N=n$ and $\textit{dim}\;M=m$. I know that I have to use the ...
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### Can I use de Rham Cohomology to prove exactness of 1-forms on the $2$ sphere?

The exercise ask to prove that for every 1-form closed on the $2$ sphere there was a function defined on the sphere such that its differential is the form. Then it asks if this function is unique. ...
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### Calculation of de Rham complex for real projective space

I've been studying de Rham cohomology and the use of the Mayer-Vietoris sequence to compute the de Rham complex for the real projective space $\mathbb P^d$. First, I divided $\mathbb P^d$ in two open ...
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### Minimal prerequisite to understand De Rham Cohomology

To understand the concept of De Rham Cohomology formally, what is the minimum background required? That is, what are the critical concepts? For example, a list might be something like: tangent ...
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### Existence of a countable locally finite cover with nonempty intersection of two adjacent elements

Let $\Omega$ be an open connected set in $\mathbb{C}$, not necessarily bounded. Does there exist a countable locally finite cover of $\Omega$ consisting of only open discs $\{ B(z_i, r_i): i\geq 1\}$ ...
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### Simplicial and De Rham Homology on Manifolds

I'm looking for a recomendable reference/source for a rigorous proof that for manifolds (with "nice enough" structure) the simlicial and De Rham (co)homologies coincide. Especially, I know that ...
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### Wedge product between relative de Rham cohomology and de Rham cohomology on a subspace

Under appropriate conditions on $U\subset X$ (what are they?), is there a well-defined wedge product $$\wedge: H^k(X,\overline U)\oplus H^l(X\setminus U)\to H^{k+l}(X,\overline U),$$ and if so, is ...
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### De Rham cohomology of $S^1$
I want to calculate $H^1_{dR}(S^1)$. I'm stuck at the beginning. I know that H^1_{dR}(S^1) = Z^1/B^1 = \frac{\{\alpha\in\Omega^1\ \vert\ X_0(\alpha(X_1)) - X_1(\alpha(X_0)) = \alpha([X_0,X_1])\}}{...