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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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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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Stokes' theorem and non-globally defined quantities

This is a question from a physicist, so please be kind.* Suppose that $M$ is an orientable smooth manifold without boundaries and $\omega$ a form of an appropriate degree such that it can be ...
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Integrating $[\alpha] \in H^1_{dR}$ along a closed curve is well-defined

Let $U := \mathbb R^2 \setminus \{0\}$ be the punctured plane and $\gamma$ the counter clockwise once traversed unit circle. Consider $$\tag{1} [α] \mapsto \int^{2 \pi}_{0} \alpha_{\gamma (t)}(\gamma'...
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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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Topological invariance of compactly supported de Rham cohomology

It is well-known that if we are given two smooth manifolds (without) boundary, whose underlying topological spaces are homotopic, then the de Rham cohomologies $H^k_{dR}$ of $M$ and $N$ are isomorphic ...
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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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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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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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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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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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Why is the de Rham complex elliptic?

I need to show that the exterior derivative $$ \mathrm{d} : \Omega^r(M) \to \Omega^{r+1}(M) $$ is an elliptic differential operator. As far as I understand it, an elliptic operator is one such that ...
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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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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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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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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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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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Finite group acting freely on a smooth manifold (De Rham cohomology)

I have a finite group $G$ acting freely on a manifold and $\pi:M\to M/G$ is the covering map. I need to show that the induced (pullback) map $\pi^*:H^*(M/G)\to H^*(M)$ is injective, where $H^*(M)=\...
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Two definitions of degree of a map

Consider these two definition of degree of a map $f : M \to N$ between closed connected and oriented differentiable manifolds of the same dimension $n$: 1)Assume that $f$ is continuous. Since $H_n(...
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How to Find a Vector Field(/Differential Form) with Div(F) = 0 but F \ne Curl(A) on R^3\(S^1x{0})

In this post, it is outlined how to find a differential $n$-form on $U_0 = \mathbb{R}^n\backslash\{\text{pt}\}$ whose exterior derivative is zero but which is not the exterior derivative of an $(n-1)$-...
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Cohomology theories and sheaf cohomology

Let $X$ be a paracompact Hausdorff topological space, $\mathcal U$ an open covering of $X$ and $\mathcal N(\mathcal U)$ the nerve of the covering (https://en.wikipedia.org/wiki/Nerve_of_a_covering). ...
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De Rham Cohomology: tridimensional space $\mathbb{R}^{3}$ without a circle

I'm a new users. I'd like to calculate the De Rham cohomology of euclidean space $\mathbb{R}^{3}$ without a circle $\mathbb{S}^{1}$. I don't have idea how to proceed! I saw the answer given to this ...
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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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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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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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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 $...
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$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 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 ...
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Why $H_{dR}^1(M) \simeq \mathbb R^n$ when $H_1(M,\mathbb Z)$ has $n$ generators?

Let $M$ be compact, connected and orientable manifold without boundary. Let $\sigma_1,\ldots, \sigma_n$ be generators of $H_1(M,\mathbb Z)$. Why is the map $I:H_{dR}^1(M) \to \mathbb R^n$ given by $$I(...
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Finite dimensionality of the “deRham cohomology” defined using $C^{k,\alpha}$ forms instead of smooth forms.

Let $M$ be a compact manifold. The deRham cohomology group $H^{^\bullet}_{DR}(M)$ is defined to be the cohomology of the deRham complex $\left({\cal C}^{\infty}(\bigwedge^{^\bullet}T^*M),d\right)$. ...
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DeRham Cohomology of punctured plane and homotopy.

If I am not wrong the homotopy type of curve in the punctured plane is dictated by winding number around the puncture. This is not the case for the doubly punctured plane (say with puncture at the ...
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Mayer-Vietoris sequence for compact support de Rham cohomology of $S^1$

This is an example that is computed in Bott and Tu's differential forms. I do not get the obvious reasoning here. Take $U,V$ covering $S^1$. Then one obtain compact support de Rham cohomology. $0\to ...
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Cup-length using de Rham cohomology

While teaching a class on de Rham cohomology, I was trying to build an exercise showing that in a space (=smooth manifold) $M$ which is the union of $k$ contractible opens, every $k$-fold product in $...
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Soft: Advantages and big results using Homology in PDEs

I'm new to the viewpoint of using (co)homological methods in the theory of PDEs. To help motivate me, I was wondering what are some significant results which have been obtained (primarily?) through ...
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Restrict differential forms that vanish along fibres to the base

I am reading Jean-Luc Brylinski's book on loop space. At the end of section 1.6 Leray Spectral Sequence, he claims without proof that Let $f: Y\to X$ be a smooth bundle of paracompact manifolds. A $...
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Regarding the proof of Poincare Duality via 5-Lemma (commutativity?)

The proof of Poincare Lemma for oriented manifolds with finite good cover (terminology of Bott, Tu) states that \begin{align} H^q (M) \simeq H^{n-q} _c (M) \end{align} where $n=\dim M$. The proof ...
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de Rham cohomology and homotopy groups of differential manifold

As is well known, a differential manifold has trivial degree-$0$ de Rham cohomology $H^0(M) = 0$ if and only if it is connected. It seems that the degree-$1$ de Rham cohomology group $H^1(M)$ being ...
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Hodge Numbers of a Calabi Yau manifold

I was going through this, where on page 6, it is mentioned that on a $2n$ dimensional Calabi-Yau manifold, $h_{(n,0)} = h_{(0,n)} = 1$. What is the reason for this? One way to prove this is by ...
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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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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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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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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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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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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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Averaging of a differential form.

Let $\omega$ be a differential form on $S^1.$ There is an action of a circle on itself by rotations ($A: S^1 \to S^1$ as $A(x)=A+x$ if we think of a circle as an additive group). Let's define the ...
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convention on exterior derivatives

What is the convention on exterior derivatives especially concerning $H^0_{dR}(M)$? Indeed, $H^0_{dR}(M) = Z^0(M)/B^0(M)$ where $B^0(M) = \Omega^0(M)\cap \text{Im}(d)$. However $d$ is not defined on $...
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What is the differential form of a composition

I would like to be able to calculate $\alpha(X_0\circ X_1)$ for $\alpha\in\Omega^1$ the set of 1-forms. Is it $X_0(\alpha(X_1))$? At least that is what I would like it to be but how can I get there?
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etale sheafification of exact forms

We know that the set of differential forms together with restrictions is a sheaf. But what happens if we take the subset of exact forms and make its etale sheafification? Do we obtain a distinct sheaf?...