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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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Sheafification of constant real sheaf on smooth manifold and sheaf of smooth functions

I have a question that has been motivated by this one. Since $H^k_{dR}(M)\cong \hat{H}^k(M;\mathbb{R}_M)$ I was wondering if the constant sheaf $\mathbb{R}_M$ was isomorphic to the sheaf of $C^{\...
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What is the field of de Rham cohomology vector space over?

On Tu's an introduction to Manifold, 2nd edition, p275 (please see the image below) It said that all the closed $k$-form and exact $k$-forms on a manifold are both vector space. I think the vector ...
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deformation of Hodge star operator and harmonic forms

Suppose $(M,g)$ is a compact Riemannian manifold, and $*_g$ is the Hodge star operator defined on the de Rham algebra $\Omega^*(M)$ with respect to the metric $g$. Let $\phi:M\to M$ be a ...
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singular cohomology and Poincaré duality

Suppose $M$ is a n-dimensional, finite type, oriented, smooth manifold. A $k$-dimensional cycle in $M$ is a pair ($S$,$\phi$), where $S$ is a compact, oriented $k$-dimensional manifold without ...
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$F:M\to N$ is surjective if $\int_M F^* \eta \ne 0$ for some $\eta \in \Omega^n(N)$

Let $M$ and $N$ be compact orientable and connected smooth $n$-manifolds and $F:M \to N$ a smooth map. Suppose $$\int_M F^* \eta \ne 0$$ for some $\eta \in \Omega^n(N)$. Then $F$ is surjective. Give ...
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Forms on vector bundles: vertically compactly supported

Definition: Let $\pi:V \rightarrow M$ be a vector bundle. $\Omega^p_{cv}(V)$ is the sections of $p$ form on $V$, such that $\pi^{-1}(K) \cap supp \, (w) $ for all $K \subseteq M$ compact. ...
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Chern class of a principal $G$ bundle for a compact Lie group $G$

This question is related to this question. The user who asked this question is not active since September. So, asking a separate question here. Let $G$ be a compact Lie group and $P\rightarrow M$ be ...
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Understanding the homotopy operator for de Rham Cohomology

This is in John Lee's Smooth Manifold 2nd Edition, pg 444 For any smooth manifold $M$, there exists a linear map $$ h:\Omega^p(M \times I ) \rightarrow \Omega^{p-1}(M)$$ such that $$ h(dw)+d(...
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Notational question on Kunneth Formula for de Rham cohomology

I got to learn the Kunneth Formula for de Rham cohomology as following. $$H^n(X\times Y)=\sum_{n=p+q} H^p(X)\otimes H^q(Y). $$ And I could find same notation from https://www.encyclopediaofmath.org/...
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Computing de Rham cohomology group $H^1(\mathbb{RP}^2\#\mathbb{RP}^2)$

I am trying to compute the de Rham cohomology group $H^p(\mathbb{RP}^{n+1}\#\mathbb{RP}^{n+1})$ and I am stuck at computing $H^1(\mathbb{RP}^2\#\mathbb{RP}^2)$. ($\#$ stand for the connected sum) ...
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Involution action on $H^3(S^1\times S^2)$

I am studying about involution action $I^*$ on de Rham cohomology group $H^3(S^1\times S^2)$ induced from an action $I\cdot (z,x)=(\overline{z},-x) $ where $S^1\times S^2\subset \mathbb{C}\times \...
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Involution action on $H^1(S^1\times S^2)$

I am studying about an action $I^*$ on a de Rham cohomology group $H^1(S^1\times S^2)$ induced from an action $I\cdot (z,x)=(\overline{z},-x) $ where $S^1\times S^2\subset \mathbb{C}\times \mathbb{R}^...
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Chern class cohomology coefficients complex/real/integral?

I am reading Chern classes from Kobayashi and Nomizu. Given a vector bundle $\pi:E\rightarrow M$ with fibre $\mathbb{C}^r$ and Group $GL(r,\mathbb{C})$ they associate for each $k\leq r$ a cohomology ...
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Reference for cup product in deRham cohomology

Given a smooth manifold $M$, we have what are called deRham cohomology groups $H^i(M,\mathbb{R})$. deRham cohomology ring $H^*(M,\mathbb{R})$ is as a set $\bigoplus_{i=0}^{\text{dim(M)}} H^i(M,\...
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$\omega$ a closed 2-form and $\bigwedge_{i=1}^n \omega \ne 0$ on a compact orientable smooth $2n$-manifold w/o boundary, $M$, then $H^2(M) \ne 0$.

Suppose $M$ is a compact orientable smooth $2n$-manifold without boundary, and let $\omega$ be a closed $2$-form such that $\bigwedge_{i=1}^n \omega_p \ne 0$ at every point $p$. Show that $H^2_{dR}(M) ...
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Computing the de Rham cohomology of $S^2$

I am trying to compute the de Rham cohomology of $S^2$ using Mayer Vietoris sequence. I considered the open cover $U$, where $U$ is the whole $S^2$ minus the north pole, and $V$, where $V$ is the ...
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How does this example from Spivak that $H_c^n(\mathbb R^n) \ne 0$?

I am not sure how this integral that is being calculated using Stoke's theorem shows that the $n$th de Rham cohomology group with compact supports of $\mathbb R^n$ is not trivial. How does the fact ...
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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 doubly punctured torus

Let $T^2=S^1\times S^1$. I'd like to know all de Rham cohomology groups of $M=T^2-\{a,b\}$ but I couldn't find a result. So I want to compute it and I'm thinking of using Mayer Vietoris sequence. I ...
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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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Computing de Rham Cohomology

I'm stuck on the following problem. Let $X=S^{n}\setminus A$, where $A$ is the union of $k\geq 1$ disks $D_{k}$. Use the Mayer-Vietoris sequence to compute the de Rham cohomology $H_{\mathrm{dR}}^{...
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De Rham cohomology of $\mathbb{RP^n}$

I have to calculate the De Rham cohomology of $\mathbb{RP^n}$ using the Mayer-Vietoris sequence. I first started by considering $\mathbb{RP^n}=S^n/\sim $ where $\sim$ is the antipodal identification. ...
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De Rham cohomology groups of $\mathbb{R}^n$

I want to show that for each $1 \le k\le n$ we have $$ H_{dR}^k(\mathbb{R}^n)=0 $$ The strategy is to construct for each $k$ a linear map $$h_k:\Omega^k(\mathbb{R}^n)\to \Omega^{k-1}(\mathbb{R}^n)$$ ...
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Is there a converse to “knowing a good cover” → “knowing cohomology”?

When looking at smooth manifolds, knowing a “nice” cover of our space enables us to calculate the De Rham cohomology via e.g. the Meyer-Vietoris sequence. For instance, $$ M := \{(x,y)\mid x^2+y^2\...
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Cohomology and left invariant 1-forms

I'm computing the de Rham cohomology of the group $SU(2)$, with $n_g$ generators, making use of the base of left invariant 1-forms $\eta^i, i = \{1, ..., n_g\}$, in order to apply the following ...
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Why intermediate map in Gysin sequence is multiplication by Euler class

I am reading Bott and Tu, Differential Form in Algebraic Topology. At page 178, they constructed Gysin sequence of Sphere Bundle. I am having trouble understanding the argument, $d_{k+1}$ is ...
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The three definitions of Cech cohomology (simplical complex vs. presheaf vs. sheaf)

I came across the following three definitions of Cech cohomology group of a topological space $X$: [Source: Munkres, Elements of Algebraic Topology, pp. 437]. The Cech cohomology group of $X$ in ...
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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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existence of de Rham complexes

I have a very basic question about the exterior derivative of differential forms and de Rham complexes. It is very basic, I know that the exterior derivative satisfies $d^2=0$. Knowing that, how is a ...
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Existence of Global Coordinates

Assume we have a smooth manifold, $M$, of dimension $n$, endowed with a Riemannian metric. (An example of interest is the case when $M$ has n=2, is orientable and compact, i.e. a compact Riemann ...
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Can every smooth scalar function $f$ on $M$ be written as $f=\operatorname{div}(X)$?

Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $X$ a vector field over it. divergence of $X$ is a real value function defined by: $$\operatorname{div}(X)=g(\nabla_{e_i}X,e_i),$$ where $\{...
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What is a generator of the de rham cohomology?

What is really meant with the word "generator" in Bott & Tu? How can a form "generate" a whole vector space of other forms?
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Double Complex with exact co-boundary operator on forms with compact support

I am reading Bott and Tu. https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf On page 140 of that book, the exactness of co-boundary operator $\delta$ is used to prove that the double complex of ...
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A direct proof of the Chern-Weil isomorphism

Given a principal $G$-bundle $P \to M := P / G$ with Lie group $G$ and associated Lie algebra $g$, the Chern-Weil homomorphism $$S^*(g)^G \to H_{DR}^*(M)$$ associated to any invariant polynomial on $...
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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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$\mathcal{H}^q$ is a locally constant presheaf

Let $\pi: E \rightarrow M$ be a fiber bundle with fiber $F$ and define the presheaf $\mathcal{H}^q(U)=H^q(\pi^{-1}(U))$, for every open subset $U \subset M$, where $H^q$ denotes the De Rham cohomology....
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$T$ is torus. $H_1(T,Z)\to \operatorname{Hom}\Omega^1(T),C)$ is injection.

Let $T=\frac{V}{\Lambda}$ be a torus where $V$ is a complex vector space of dimension $n$ and $\Lambda$ is a rank $2n$ lattice in $V$. $\Omega^1(T)$ is the space of holomorphic 1-forms of $T$. $dim_C(\...
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Intuition about calculation in Mayer-Vietoris sequence

I'm interested in calculating de Rham cohomology groups of torus $\mathbb{T}^{2}=\mathbb{S}^{1}\times \mathbb{S}^{1}$, my approach is Mayer-Vietoris Sequence, but i don't know find open sets $U$ and $...
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De Rham cohomology of $\mathbb{R}^3$ minus a circumference

Let $M=\mathbb{R}^3-\{\left(x,y,z \right) \ | \ x^2+y^2=1 ,\ \ z=0 \}$. I'd like to calculate its De Rham cohomology. I've tried to use Mayer Vietoris sequence applied with open sets $U=M-\{x=0 \ \ y=...
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Do you need a differentiable homotopy for the Poincaré Lemma

The Poincaré Lemma (Amann-Escher XI.3.11) states If $X$ is contractible, then every closed differential form on $X$ is exact, where contractible means that the identity on $X$ is null-homotic. ...
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De Rham cohomology of punctured manifold

I'm trying to solve the following problem (Lee's Intro to Smooth Manifolds, 17-6): Let $M$ be a connected smooth manifold of dimension $n \geq 3$. For any $x \in M$ and $0 \leq p \leq n-2$, prove ...
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If $\omega \in \Omega^{n-1}(M)$ then $d_p\omega = 0$ at some $p \in M$.

Let $M$ be a compact, oriented, smooth manifold of dimension $n$. I have to show that if $ \omega \in \Omega^{n-1}(M)$ then $d_p\omega = 0$ at some $p \in M$. My first attempt was to use the ...
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why is the exterior derivative natural?

In the study of manifolds up to a certain point, all the definitions feel natural in the sense that there's no other way to define them. I'm thinking of concepts like tangent vectors, one-forms, ...
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Integer coefficients in de Rham Cohomology

It can be shown that $H_{dR}^*(X,\mathbb R)$ and $H^*(X,\mathbb R)$ (singular cohomology) are isomorphic for smooth manifolds. I was told that under that isomorphism closed differential forms whose ...
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$H^1(S^1)\cong \mathbb{R}$ from definition

This is a problem from Jänich's Vector Analysis: Prove directly from the definition that $[\omega] \rightarrow \int_{S^1}\omega$ defines an isomorphism $H^1(S^1)\cong \mathbb{R}$, and go on to show ...
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Question about the relation between two different ways of definition the degree of map

I am learning the Riemann Surfaces and I have a question when I read the appendix A.1.5. of textbook A course Complex analysis and Riemann surfaces by W.Schlag. One way to define the degree of the ...
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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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Terminology regarding deRham cohomology “group”

This is quite an elementary question, but I'm confused over terminology and whether $H_{dR}^k(M)$ is a module, or vector space (this confusion is exacerbated when it's often called the de Rham "group")...
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The Dimension of a Quotient Vector Space

Consider $X= \mathbb{R}^2-\{(0,0),(1,0),(2,0),(3,0)\}.$ Let $V$ be the vector space of irrotational vector fields over $X.$ Let $W$ be the vector space of conservative vector fields over $X.$ What is ...
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Question about differential forms and delta functions

EDIT: Related to "Derivative of a logarithm and Dirac delta function", "2-dimensional delta function (complex plane)" and "How to define a delta function on complex plane?". In a physics paper by ...