# Questions tagged [de-rham-cohomology]

This a cohomology theory for smooth manifolds, where the (co)chain complex is defined by differential forms on a smooth manifold with differential given by exterior derivative. Then $n^{th}$ de Rham cohomology group is the quotient "closed $n$-forms/exact $n$-forms". Use in conjunction with other algebraic topology and differential geometry related tags if necessary.

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### Mayer-Vietoris for smooth manifold with boundary

Disclaimer: I am aware that this question is entirely moot, since any smooth manifold with boundary is homotopic to its interior! Nonetheless, I am still interested in learning what goes wrong if we ...
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### Compute the induced homomorphism on deRham cohomology

Consider the smooth map on the torus to itself $f: S^1 \times S^1 \rightarrow S^1 \times S^1$ defined by $f(z_1, z_2) = (z_1^2z_2, z_1^{-1}z_2)$ (here we identify $S^1$ as the unit circle on the ...
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### How to prove the existence of differential forms on a manifold using de Rham cohomology?

Let $S^3$ be the 3-sphere, and $\Sigma$ be a 2-dimension manifold. Let $\omega$ be a 2-form on $\Sigma$. $f:S^3\rightarrow \Sigma$ is a $C^{\infty}$ map.Then there is a 1-form $\alpha$ on $S^3$ such ...
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### What is "torsion" in the context of cohomology, and why is it important?

I searched for some answers, but most answers discussed the meaning of torsion, instead of its definition. Not knowing how the torsion is defined (in cohomology) I couldn't understand those answers at ...
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### Question about the Euler class

Let $M$ be a compact orientable $n$ dimensional smooth manifold without boundary, and $e\in H^n_{dR}(M)$ denote the Euler class of the tangent bundle $TM$. We have that \begin{align*} \int_Me=\chi(M) \...
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### Gauss-Manin connection of a family over a formal disk

Let $X$ be a smooth proper scheme over $\mathbb{C}[[t]]$ and $X_0$ its special fiber. The cohomology $H^*_{dR} (X/\mathbb{C}[[t]])$ admit the Gauss-Manin connection which is determined by the action ...
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### Contravariant functor $\Omega^*$ from category of smooth manifolds to commutative differential graded algebras

I thought I understood the Mayer-Vietoris Sequence in the context of De Rham cohomology, but I have realized that I have taken something for granted in the set up of the proof, and I can't quite ...
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### Big confusion about cohomology of complex manifold

First, I never learned complex manifold properly (I am more familiar with complex variety) hence the questions will be very elementary. I apologise for this. Let $X$ be a complex smooth manifold of ...
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### Do we still use vector potential in Maxwell's equations?

In Maxwell's equations, since field $\mathbf{B}$ is divergence free, we can find vector potential $\mathbf{A}$ such that $\mathbf{B}=\nabla\times\mathbf{A}$. However, this identity holds only when ...
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### Understanding the calculation of Rham’s first cohomology group of the puntured plane

I have a question: How is the first Rham cohomology group computed? I revised the answer in 2, but I don't understand the proof. Let M be the puntured plane and $\alpha \in \Omega^1(M)$ be a closed ...
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### First cohomology group of Kähler-Einstein manifolds [closed]

I have read that the first cohomology group of a Kahler-Einstein manifold of dimension four, with positive scalar curvature, is zero. But I cannot find a proof. How can I prove it? Where can I find a ...
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### Counterexamples to "irrotational implies potential"

I'm trying to figure out some counterexamples of the following: If $F$ is an irrotational vector field then there exists some $U$ such that $F = \text{grad}(U)$ (or more generally, using the language ...
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### On a certain "anticurl" operator

I recently found myself curious what explicit formula I would get if I traced through the de Rham cohomology proof that if $\mathbf{F}$ is a vector field defined on all of $\mathbb{R}^3$ which has ...
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### Is there a smooth global section from projective space to the sphere?

I'm having a hard time trying to find (if there is any) global section from $\mathbb{RP}^n$ to the sphere $\mathbb{S}^n$ or $\mathbb{R}^{n+1}$ (global sections of the the natural projection map $\pi$ ...
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### Computing the Euler Characteristic of $\mathbb{CP}^2 \# \mathbb{CP}^2$

I'm very new at algebraic topology. I'm trying to compute the Euler characteristic of $\mathbb{CP}^2 \# \mathbb{CP}^2$ by using the following known facts. ($\mathbb{CP}^2 \# \mathbb{CP}^2$ represents ...
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### Is it possible to study de Rham Cohomology of surfaces through Riemann surfaces?

I am studying de Rham Cohomology of surfaces. I am following the book "An Introduction to Manifolds" by Loring Tu. I found that it uses the "Mayer Veitoris Sequence" and notions ...
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### What is the product on Kunneth formula for deRham cohomology?

The Künneth theorem states that for every k>0, $$H_{dR}^k(M \times N) = \bigoplus_{i+j=k} H_{dR}^i (M) \otimes H_{dR}^j (N).$$ I would just like to know: what is this product inside the sum? Is it ...
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### Hochschild cohomology and de Rham cohomology

I know that given a real finite dimensional associative algebra $A$, its automorphism group $\text{Aut}(A)$ is a Lie group. Is there some relation between the de Rham cohomology of this Lie group and ...
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Let $S$ be a compact oriented Riemannian surface. And let $\mathcal{H}^1(S)$ be the space of the harmonic $1$-forms of $S$. I’m trying to prove that there is a linear isomorphism: $$H^1_{dR}(S)\to \... 5 votes 1 answer 81 views ### Is the cohomology of a pushout of open sets a pullback of cohomologies? I recently found an interesting question, and I'd like to ask the co-question. I will differ from what is linked by using de Rham cohomology, since its the easiest way for me to articulate my ... 3 votes 2 answers 351 views ### Computing the De Rham Cohomology Group of Connected Sum H_{\rm dR}^p(M_1 \# M_2) I'm reading Lee's Introduction to Smooth Manifolds. I have a question about problem 17-7. Problem 17-7 Let M_1, M_2 be connected smooth manifolds of dimension n\geq3, and let M_1\# M_2 denote ... 0 votes 0 answers 65 views ### De Rham cohomology of \mathbb{CP}^n by Mayer-Vietoris sequences? I wonder if one can compute the De Rham cohomology of \mathbb{CP}^n only via Mayer-Vietoris sequences, here I mean we make use only of the usual open covering of \mathbb{CP}^n, i.e., U_i = \{[z_0:... 0 votes 1 answer 64 views ### De Rham cohomology of connected components I am studying a bit about De Rham cohomology and, in the reference I am using, it says that is clear that: If M is a compact, orientable and differenciable manifold of dimension n \geq 1 with ... 2 votes 1 answer 188 views ### Calculate the de Rham cohomology of S^1 \times S^2 For our homework sheet it says: calculate the de Rham cohomology of M = S^1 \times S^2 or check a book if you are unsure. And then it continues. Since I didn't find the calculation in a book I ... 0 votes 0 answers 22 views ### Top de rham degree of the total space of a vector bundle By homotopy invariance of the de Rham cohomology and the fact that the total space of every smooth vector bundle E\to M is homotopy equivalent to its base space M (M is a smooth manifold), we ... 2 votes 1 answer 128 views ### interior product and cohomology We know that we can define the de-Rham cohomology because d^2=0. However, we also have the property \iota_X \iota_X=0 for interior product, so I am wondering can we also define a cohomology for \... 1 vote 0 answers 39 views ### Does an equivariant map (/deformation retract) which induces an isomorphism in cohomology also induce an isomorphism in equivariant cohomology? Let G be a compact Lie group acting smoothly on two manifolds M and N and suppose we have an equivariant map f: M \rightarrow N which induces an isomorphism in cohomology f^*: H^*(N) \... 2 votes 1 answer 83 views ### Generator of H^1(S^1) via integration of a bump 1-form on S^1 I have a statement in Bott & Tu's Differential form in Algebraic topology (p.36) that I can't understand. We say previously that a generator of H^1(S^1) is a bump 1-form on S^1 which gives ... 0 votes 1 answer 85 views ### Is there a manifold M\subset\mathbb{R}^3 s.t. H^k(M)=\mathbb{R} for k=1,2,3? (deRham cohomology groups) I was thinking the following: First, note that \mathbb{S}^1\subset\mathbb{R}^2 and we have the deRham cohomology groups H^{k}(\mathbb{S}^1)=\mathbb{R} for 0\leq k<2 and 0 otherwise, but \... 3 votes 0 answers 45 views ### For a geom. integral curve X/k of char p, do the constants \mathrm{ker}\left[K(X) \to \Omega^1_{X/k}\otimes K(X)\right] equal K(X)^{(p)}? The motivations for this question are related to Constants of universal derivation on an R-algebra A under localization (d: S^{-1}A \to S^{-1}\Omega_{A/R}) Let X be a geometrically integral ... 6 votes 1 answer 136 views ### Is there a topological point of view for Liouville's theorem about elliptic functions? Good evening! One of Liouville's theorems about elliptic functions states that there is no such function that has only one pole of order 1. This result is very well known and easily proven using the ... 0 votes 0 answers 60 views ### Exterior Algebra with Jets instead of Forms? For a smooth, n-dimensional manifold M over \mathbb{R}, I would like to know if sections of the dual space of the order-k jet bundle sit within a commutative differential graded algebra ... 4 votes 0 answers 154 views ### Different comparisons between de Rham and singular cohomology Let X be a smooth manifold. I am aware of two comparison isomorphisms H^*_{dR}(X,\mathbb{R}) \rightarrow H^*_{sing}(X,\mathbb{R}) between de Rham cohomology and singular cohomology (with real ... 1 vote 1 answer 115 views ### Poincare duality for algebraic de Rham cohomology with integrable connection coefficients I am reading "The Gauss-Manin Connection and Tannaka Duality" (here is the link to the paper). I am specifically interested in the proof of Proposition 2.2. In this proof, the authors use ... 1 vote 0 answers 25 views ### Confusion about the following diagram I was reading the book here, and in 2.1.11, there is a diagram which involves an arrow looking like this:$$ H^0_{\mathrm{dR}}(M) \hookrightarrow \Omega^0(M)/\Omega^0_{\text{cl}}(M)_{\mathbb{Z}}  ...
Notation: for $X$ a smooth manifold, let $T^n(X)$ denote the space of (global, smooth) covariant rank-$n$ tensors on $X$, i.e. sections of $(T^*X)^{\otimes n}$. Let $X\xrightarrow{Q}Y$ be a smooth, ...