# 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".

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### algebraic de Rham cohomology of blowup of relative line

Let $$T = \mathbb{A}^1_k,\,\,\, Y = \mathbb{P}^1_T,\,\,\, X = \mathscr{B}(Y),$$ the blowup of $Y$ at a point. I am trying to compute the de Rham cohomology $H^1_{dR}(X/T)$, but I could use some help. ...
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### Tu's An Introduction to Manifolds - Section 26.2 Cohomology of a circle, tabular form.

I'm trying to understand how to use the Mayer-Vietoris sequence to compute Cohomologies. There's a small chapter in Tu's Introduction to Manifolds explaining the basics, with some basic theory. More ...
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### De Rham cohomology of $S^1$ with compact supports (Bott/Tu)

This is a question about Example 2.9, in Bott/Tu - Differential Forms in Algebraic Topology. Consider the decomposition of $S^1=U\cup V$ by two open sets, as in the figure above. Then both $U$ and $V$...
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### First sheaf cohomology of closed 1-forms on a homogeneous space

Given a smooth quasi-projective variety $X$ over $\mathbb C$, is there a good way to compute the first sheaf cohomologiy $H^1(X,\Omega_{X,cl}^1)$ of closed 1-forms $\Omega_{X,cl}^1$? What I'm mostly ...
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### Poincare Duality and Integration Isomorphism

Corollary 10.27 in Jeffrey Lee's book "Manifolds and Differential Geometry" states that If $M$ is a connected oriented $n$-manifold with finite good cover, then $H^n_c (M) \simeq \mathbb{R}$. ...
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### symplectic form, tautological 1-form and DeRham cohomology

There's a question in Lee's Introduction to Smooth Manifold that asks to prove that the Grassmannian product of a symplectic form is not exact. However, isn't this incorrect if there exists a ...
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### Differential forms and hairy ball theorem

Take the $2$-form $\omega = \sin\phi \,\mathrm d \phi \wedge \mathrm d \theta$ in spherical coordinates, defined on the portion of $\mathbb R^3$ outside of the unit sphere. Clearly, $\omega$ is closed ...
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### Is there an inverse of the Stokes' Theorem for manifolds with boundary?

I would like to consider this question When does a null integral implies that a form is exact? (also related to Top deRham cohomology group of a compact orientable manifold is 1-dimensional), but for ...
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### Cohomology of $T^{2} \times \mathbb{R}^{2}$

I need to find the cohomology groups $H^{k}_{dR}$ of $T^{2} \times \mathbb{R}^{2}$ (where by $T^{2}$ I mean the two-torus). I would like to use the Mayer Vietoris sequence. Any advice on which open ...
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### Resolution of the constant sheaf $\mathbb C$.

The differential-graded algebra of complex-valued smooth differential operators $\mathcal{A}_{X, \mathbb C}^{\bullet}$ on an $n$-dimensional complex manifold $X$ (real dimension $=2n$) is acyclic ...
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### Doubt about obtaining integral cohomology form from a rational cohomology form

The above text is from Griffiths and Harris. $M=V/\Lambda$ is a complex tori where $V$ is a $n$-dimensional complex vector space and $\Lambda$ is a lattice isomorphic to $\mathbb{Z}^{2n}$ in $V$. We ...
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### Computing the compactly supported de-Rham cohomolgy $H^{1}_{c}(\mathbb{R})$

I am asked to show that $H^{1}_{c}(\mathbb{R})\cong\mathbb{R}$. In coordinates, a compactly supported 1-form on $\mathbb{R}$ can be written as $\omega=f(x)\mathrm{d}x$, where $f(x)$ is a compactly ...
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### Fiber of direct image of relative de Rham complex in non-proper case

Let $f: X \rightarrow B$ be a smooth family of complex varieties ($X$ and $B$ are also smooth). If $f$ is proper, then the direct image of the relative de Rham complex, $Rf_*\Omega_{X/B}^{\bullet}$,...
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### Proving the de Rham cohomology of $M$ and $M \times \mathbb R$ are isomorphic without homotopy

Given a smooth manifold $M$, is there an elementary way of showing $H^k_{dR} (M) \cong H^k_{dR} (M \times \mathbb R)$, i.e. straight from the definitions of cohomology being the quotient of closed ...
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### Definition of differential forms from derivations

I was skimming through Taylor's book "Several complex variables with connections to algebraic geometry and Lie groups" (first chapter was the best introduction i have ever read!), glimpsed to section ...
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### De Rham's cohomology groups by definition

I have to find the De Rham's cohomology groups of $X$, where: a)$X=S^1\times S^1$. b) $X=\mathbb{R}^3\setminus \mathbb{R}$. c) $X=\mathbb{R}^3\setminus S^1$. (d) $X=\mathbb{R}^3\setminus (L_1\cup C)$...
If the Lie algebra $\mathfrak{g}$ can be realized as the tangent space of a compact Lie group $G$, then all the possible central extensions of $\mathfrak{g}$ are in one to one correspondence which the ...