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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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Properties of the de Rham complex for $\mathbb{R}^{3}$

I wasn't able to find a construction of de Rham complex for $\mathbb{R}^{3}$ using de Rham theorem. This is my attempt, in which I have some uncertainties. Consider $$ \Omega^{0}(\mathbb{R}^{3},\...
Matthew Willow's user avatar
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Čech cohomolog of a good cover of the real projective plane

I was reading the Bott/Tu book and trying compute the Čech cohomology of a good cover of the real projective plane (exercise 9.10). The nerve of the cover is dipicted as below: we glue the opposite ...
Xipan Xiao's user avatar
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Relation between de-Rham cohomologys on a three-manifold $Y$ and the cylinder $Y\times \mathbb{R}$

Consider the deRham complex of a compact three-manifold $Y$ and the associated deRham cohomology group $H_{dR}^1(Y; \mathbb{R})$. Say there is a principal $G$-bundle $P$ over $Y$ such that the 1-forms ...
Simp's user avatar
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$L^2(\mathbb{R})$ 1-form not differential of $L^2(\mathbb{R})$.

In my attempt to solve following exercise: Construct a smooth 1-form on $\mathbb{R}$ with $\int |\psi|^2 dt < \infty$ for which there exists no function $f$ on $\mathbb{R}$ such that $\int |f|^2 ...
Pastudent's user avatar
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Induced morphism in Čech cohomology

Let $X, Y$ be topological spaces. If $F$ is a sheaf on $Y$ we denote by $f^{-1}F$ the pullback sheaf on $X$. I know that there is a morphism induced between the sheaf cohomology groups $$ H^q(Y,F)\...
Juan MF's user avatar
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Constructing an isomorphism $H^{k}(M_1-W_1)\oplus H^{k}(M_2-W_2)\rightarrow H^{k}(M_1\# M_2)$

I'm back again. Let $M_1$ and $M_2$ be compact, orientable, connected smooth manifolds. In trying to understand the first question of this question, I want to construct an isomorphism $H^{k}(M_1-W_1)\...
A Name's user avatar
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An example of two manifolds whose deRham cohomology groups are isomorphic but whose deRham cohomology rings are not.

The full question is as follows: give an example of two manifolds $M$ and $N$ such that $H^k(M)\cong H^k(N)$ but the cohomology rings $H^*(M)$ and $H^*(N)$ are not isomorphic. Now, I was given the ...
A Name's user avatar
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De Rham Cohomology of $M\times S^n$

I was recently asked to prove that $$H^k(M\times S^n)\cong H^k(M)\oplus H^{n-k}(M)$$ for all $k\in\mathbb{Z}$ and $M$ a compact $C^\infty$ manifold without boundary. At the time, we had nothing ...
Dowdow's user avatar
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The natural maps $H^k(X,\mathbb{C})\to H^k(X,\mathcal{O}_X)$ and $H^k(X,\mathbb{C})\to H^{0,k}(X)$ coincide. (Lemma 3.3.1 Huybrechts)

The proof makes use of this diagram. Using it, one can describe $H^k(X,\mathbb{C})\to H^k(X,\mathcal{O}_X)$ induced by the inclusion $\mathbb{C}\subset \mathcal{O}_X$ in terms of explicit ...
領域展開's user avatar
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de Rham cohomology is isomorphic to the harmonic forms on compact Kähler manifolds [duplicate]

Corollary 3.2.12 in Huybrechts's Complex Geometry states that the de Rham cohomology on compact Kähler manifolds can be decomposed in terms of the Dolbeault cohomology: $$ H^k(X, \mathbb{C}) \cong \...
領域展開's user avatar
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The symbol of de Rham cohomology on Stacks Project

In the chapter of de Rham cohomology on Stack Project, there is a symbol $H^{i}(R\Gamma(X, \Omega_{X / S}^{\bullet}))$. What does $R\Gamma$ mean?
jhzg's user avatar
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Integral of 4-form on $\mathbb{C}P^2$

Let $U = \mathbb{C}^2 \simeq \mathbb{R}^4$ and $U \subset \mathbb{C}P^2$ by $(u, v) \mapsto [1:u:v]$. Consider the differential form $$\eta = \frac{r^2d\theta + s^2d\psi}{1+r^2+s^2}$$ on $U$, with $(u,...
Jahi02's user avatar
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Independence of an integral of a differential form on a sphere

Let $n \geq 2$ and $f: S^{2n-1} \rightarrow S^{n}$ be a smooth map. Let $\omega$ be an volume form on $S^n$ such that $\int_{S^n}w = 1$. We know that there exists a (n-1) form $\beta \in \Lambda^{n-1}(...
Jahi02's user avatar
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Twisted Poincaré duality reduces to usual one when manifold is oriented

Theorem 12.15 of Differential Forms in Algebraic Topology by Bott and Tu states that when $M$ is a smooth of dimension $n$ and $\mathfrak U$ is a a good cover of $M$ satisfying the condition $(*)$, ...
JerryCastilla's user avatar
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1 answer
77 views

Ring structure on relative de Rham cohomology

In Bott-Tu, for a smooth map $f:S\to M$ between two manifolds, they define a complex $\Omega^*(f)=\bigoplus_{q\geq 0} \Omega^q(f)$ by $\Omega^q(f)=\Omega^q(M)\oplus \Omega^{q-1}(S)$ with differential $...
blancket's user avatar
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A proof of a form of the Poincare Lemma:

If $\omega$ is a closed $k-$current on the interior of $S_n:=\{x \in \Bbb{R}^n: 0 \leq x_i \leq 1, 0 \leq x_1+...+x_n \leq 1\}.$ Then $\omega = d \eta$ where $\eta$ is an extendible $(k-1)-$current or ...
homosapien's user avatar
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How to prove the de Rham complex of sheaves of modules.

My question: Let $X$ be a topological space. Let $\mathcal{A} \rightarrow \mathcal{B}$ be a homomorphism of sheaves of rings. Denote $d: \mathcal{B} \rightarrow \Omega_{\mathcal{B} / \mathcal{A}}$ ...
jhzg's user avatar
  • 301
2 votes
1 answer
119 views

Prove that $d_3d_3^*+d_3^*d_3=-\nabla^2$

Consider the geometry in $\mathbb R^3$, define $$d_3=dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}+dz\frac{\partial}{\partial z}.$$ We then define the Hodge star operator $*_3:\Omega^p(\...
Ho-Oh's user avatar
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Steenrod squares and higher cup products for differential forms?

I am physicist, so I am sorry if I am not too rigorous in the following. I have two (closely related I guess) questions: Let me consider a triangulated manifold $M$ and its simplicial cohomology. ...
Weyl's user avatar
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0 answers
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How to compute degree of $f: S^1\to S^1$ defined as $e^{ix}\mapsto e^{iax}$?

How to compute degree of $f: S^1\to S^1$ defined as $e^{ix}\mapsto e^{iax}$? It is known that $f$ induces a vector space isomorphism $H^1(f): \mathbb R\to \mathbb R$ which is multiplication by a ...
Koro's user avatar
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Cohomology defined by a different map is isomorphic to de Rham cohomology

Let $d$ be the derivative map for de Rham cohomology. If we define $d_f=m_{-f}\circ d\circ m_f$ where $m_f(\omega)=e^f\omega$ for a smooth function $f:M\to\mathbb{R}$. Notice that $d_f^2=0$. How do we ...
Michael Wang-Wakamatsu's user avatar
7 votes
1 answer
206 views

How is the sheafy De Rham cohomology functorial?

$\newcommand{\T}{\mathscr{T}}\newcommand{\C}{\mathscr{C}^\infty}$I've been enjoying Iversen's book on sheaf cohomology. He briefly mentions De Rham cohomology and a sheafy perspective on it but he ...
FShrike's user avatar
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De Rham isomorphism for relative and compactly supported cohomology

I am currently reading up on cohomology with compact supports, and am looking for a reference as to whether the De Rham isomorphism $H_\textrm{DR}^*(M)\simeq H^*(M;\mathbb R)$ exists for the compactly ...
peabrainiac's user avatar
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Property of functions defining the set of Kaehler metrics in the same cohomology class as a Kaehler form on a compact complex manifold

Suppose $(M, J)$ is a compact complex manifold of complex dimension $n$ and there exists a Kaehler metric $\Omega$ on $M$. By the global $\partial \bar\partial$ lemma, any Kaehler metric in the same ...
rosecabbage's user avatar
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3 votes
1 answer
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Calculate $H^*(M - \{ p \})$ in terms of $H^*$ for $M$ a closed manifold

This is from my graduate level differential geometry class. Let $M$ be a closed manifold. I am trying to calculate $H^*(M - \{ p \})$ in terms of $H^*$. Here is what I have so far: We know from ...
Squirrel-Power's user avatar
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1 answer
108 views

De rham cohomology over complex manifolds

I am studying sheaf cohomology of complex manifolds and, while reading some proof about Dolbeault cohomology, I realized that there is a $\bar{\partial}$-Poincaré Lemma which gives us the local ...
user720386's user avatar
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incidence coefficients in homological integration

In the theory of CW complexes, the incidence coefficient between a cell $\sigma$ of dimension $k$ and a cell $\tau$ of dimension $k - 1$ of a complex is defined as the topological degree of the map $\...
Daniel Shapero's user avatar
1 vote
1 answer
155 views

How can de Rham cohomology find obstructions?

A differential form is closed iff its exterior derivative is $0$. A differential $k$-form $w$ is exact iff there exists a differential $(k-1)$-form $\eta$ so that $\hbox{d}\eta = \omega$. Every exact ...
étale-cohomology's user avatar
1 vote
1 answer
84 views

Is it true that every top de Rham cohomology class of a symplectic manifold $(M,\omega)$ is of the form $[f\omega^n]$?

Given a symplectic manifold $(M^{2n},\omega)$ is it always true that every top de Rham cohomology class $[\alpha]\in H^{2n}(M)$ is of the form $[\alpha]=[f\omega^n]$ for some smooth function $f\in C^{\...
Uncool's user avatar
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Holomorphic 1-forms as a subspace of the first de Rham cohomology group of a surface

If we fix a complex structure $c$ on a closed oriented surface $S$, then the space of holomorphic 1-forms $\Omega^1(S,c)$ has complex dimension equal to the genus $g$ of $S$ and - since they are all ...
Christian's user avatar
1 vote
0 answers
72 views

Compactly supported de Rham cohomology groups of $M\times \mathbb{R}^{n}$

My question is regarding a proof of the fact that $H_{c}^{k+n}(M\times \mathbb{R}^{n})\approx H_{c}^{k}(M)$ whenever $M$ is an oriented $m$-manifold whose compactly supported de Rham cohomology groups ...
dwhydtea's user avatar
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1 answer
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On the conclusion of the (compact) de Rham cohomology of $\mathbb{R}$

My question is on the compact cohomology of $\mathbb{R}$. It is easy to see that $H_c^0(\mathbb{R}) = 0$ because we can't have constant functions on compact support (except for the trivial one). Now ...
user57's user avatar
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2 votes
0 answers
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Sequence of Inclusions in the Definition of Mayer-Vietoris Sequence is unclear

I'm currently reading Bott and Tu's "Differential Forms in Algebraic Topology". The authors introduce the Mayer-Vietoris Sequence as follows: Suppose $M = U \cup V$, with $U,V$ open. Then ...
NG_'s user avatar
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4 votes
1 answer
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Is the de Rham complex independent of the smooth structure?

The de Rham complex of a smooth manifold $M$ of dimension $n$ is the complex of differential forms $$\cdots\rightarrow 0\rightarrow\Omega^0(M)\rightarrow\Omega^1(M)\rightarrow\cdots\rightarrow\Omega^n(...
Martin Frenzel's user avatar
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192 views

I have a question in hodge decomposition and ker(δ)/im(δ)-an example

I was reading the textbook "Hodge Decomposition A Method for Solving Boundary Value Problems" by Günter Schwarz. I wanted to find a simple example of the “hodge-morrey-friedrich ...
Junyeol Choi's user avatar
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0 answers
28 views

How do I show the connecting homomorphism is well defined?

Is the connecting homomorphism $ d^* : H^k(\mathcal C) \to H^{k+1}(\mathcal A) $ well defined? $0 \to \mathcal A \to \mathcal B \to \mathcal C \to 0$ is short exact. $\mathcal A$, $ \mathcal C$ are ...
oxedex's user avatar
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0 answers
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Integral classes de Rham cohomology and Čech cohomology

Setup: A smooth manifold $M$, and a locally finite contractible open cover $\mathscr{U}$ with a subordinate partition of unity $\{h_i\}$. Define a Čech cohomology $H^n(M, \mathscr{U})$ in the usual ...
A.D.'s user avatar
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1 vote
0 answers
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Self-intersection of diagonal vs. alternating sum of Betti numbers.

Let $X$ be a compact, oriented, connected manifold. I recently learned about a cohomological version of Lefschetz' fixed point theorem, which states: If $f: X \to X$ is a continuous self-map, then $$\...
red_trumpet's user avatar
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2 votes
0 answers
28 views

Is there a formula for d_2 of the de Rham spectral sequence of a fiber bundle with a connection?

The de Rham spectral sequence of a fiber bundle with a connection was studied by Hattori in a paper of essentially that as title. He works out E_0,d_0 and E_1, d_1 and E_2 but not d_2. Has it been ...
jim stasheff's user avatar
2 votes
1 answer
154 views

Example 24.2 (De Rham Cohomology of a Circle) - Tu's Introduction to Manifolds

A couple of things in the following examples that are not quite clear to me. I'll write the full proof and my question in the middle. Let $S^1$ be the unit circle in the $xy$ plane. By proposition 24....
user8469759's user avatar
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1 vote
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Zeroth cohomology of manifold with $\mathbb Z_2$ coefficients [duplicate]

Let $M$ be a smooth manifold. Then the zeroth de Rham cohomology $$H_{\text{dR}}^0(M,\mathbb R)\cong \mathbb R^{\pi_0(M)}$$ counts the number of connected components of $M$. What does the 0-th ...
ModularGymnastics's user avatar
2 votes
0 answers
47 views

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 ...
ArchimedeezNuts's user avatar
5 votes
1 answer
88 views

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 ...
mathlover314's user avatar
2 votes
0 answers
87 views

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 ...
Amemiya's user avatar
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3 votes
1 answer
246 views

Second De Rham Cohomology of $\mathbb{P}^2_\mathbb{R}$

I'm trying to show that $H^2(\mathbb{P}^2_\mathbb{R})=0$ by pulling back closed 2-forms $\omega$ on $\mathbb{P}^2_\mathbb{R}$ to $S^2$ using the fact that $\pi^*\omega$ is exact (where $\pi:S^2\...
LiminalSpace's user avatar
1 vote
1 answer
98 views

De Rham Cohomology of $\mathbb{P}_\mathbb{R}^2$

I'm trying to show that every closed 1-form on $\mathbb{P}_\mathbb{R}^2$ is exact. Towards this end, let $\alpha:S^2\longrightarrow S^2$ be the antipodal map, and $\pi:S^2\longrightarrow \mathbb{P}_\...
LiminalSpace's user avatar
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115 views

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 ...
Youngsub Yoon's user avatar
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1 answer
117 views

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) \...
Chris's user avatar
  • 3,431
2 votes
1 answer
49 views

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 ...
Chris's user avatar
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0 votes
0 answers
116 views

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 ...
Molang's user avatar
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