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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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Cup and wedge product in singular and de Rham cohomology

De Rham's theorem asserts that the map $I: H_{dR}^p(M) \to H_{sing}^p(M, \mathbb{R})$ defined as $$I(\omega)= [\sigma^p] \mapsto\int_{\sigma^p}\omega $$ is an isomorphism ($\sigma^p \in [\sigma^p] $ ...
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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 $. ...
6
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1answer
319 views

Question Regarding the Coordinate Independent Form of the Exterior Derivative

The coordinate version of the exterior derivative $d:\Omega^k(M)\to \Omega^{k+1}(M)$ of differential forms of a $C^\infty$ manifold $M$ can be expressed on a form $$ \omega=fdx^1\wedge\cdots\wedge dx^...
6
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1answer
249 views

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 ...
6
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1answer
753 views

Homotopy invariance of de Rham cohomology

Let $M,N$ be smooth manifolds which are homotopy equivalent i.e., there exists smooth maps $F:M\rightarrow N$ and $G:N\rightarrow M$ such that $F\circ G$ is homotopic to identity map on $N$ and $G\...
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1answer
51 views

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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2answers
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The Converse of Poincare Lemma

The Poincare lemma states that contractibility implies triviality of the de-Rham cohomology group. Does the converse still true? If the de-Rham cohomology is trivial, then the manifold is contractible?...
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169 views

How does one introduce characteristic classes

How do you introduce or how are you introduced to characteristic classes. I am assuming the student is comfortable with principal bundles and connections on principal bundles.
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1answer
91 views

$F:M\to N$ 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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1answer
232 views

Chern-Weil homomorphism and Chern/Pontryagin/Euler class

I am reading Chapter on characteristic classes from Foundations of Differential geometry by Kobayashi and Nomizu. This chapter starts with concept of Chern-Weil homomorphism. Given a Lie algebra $G$ ...
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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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1answer
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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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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 ...
4
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1answer
89 views

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 ...
3
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1answer
111 views

Two forms related by an automorphism are in the same cohomology class?

Let $f: M \to M$ define an automorphism on the smooth manifold M. Given a differential form $\omega \in \Omega^k$ is it true that the de Rham cohomology class of $\omega$ and $f^*\omega$ are the ...
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1answer
68 views

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 ...
3
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1answer
95 views

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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1answer
136 views

One form of poincare duality imply the other?

I saw in my differential geometry class that Poincare duality states: $H_{DR}^k(X,\mathbb{R}) \times H_{DR}^{n - k}(X,\mathbb{R}) \rightarrow \mathbb{R}$ given by $([v],[r]) \mapsto \int_{X} n \...
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1answer
199 views

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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1answer
62 views

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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0answers
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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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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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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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0answers
36 views

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 ...
3
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2answers
462 views

De Rham cohomology groups of projective real space

I would like to calculate the de De Rham cohomology groups of projective real space $\mathbb{RP}^{n}$. Well, i know all groups of De Rham cohomology os $n$-sphere $\mathbb{S}^{n}$ and that the map $\...
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0answers
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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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2answers
553 views

If a smooth, compact, connected manifold $M$ is not orientable then its top De Rham cohmology is zero

I know how to prove that if a (smooth, compact, connected) manifold of dimension $n$ is orientable then $H^n_{dR}(M) = \mathbb{R}.$ I was wondering why the converse is true, i.e., if the top comology ...
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2answers
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If $A$ and $B$ are homeomorphic proper closed subsets of $\mathbb{R}^n$, do their complements have the same homology?

Let $f_1,f_2:[0,1]\to S^3$, $g:M\to S^3$ and $h:\mathbb{R}P^2\to S^3$ be inyective maps, where $M$ is the Möbius strip. Assume that $\mathrm{Im}f_1\cap \mathrm{Im}f_2=\emptyset$. I want to compute $...
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1answer
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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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1answer
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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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1answer
50 views

$f:N\to M$ with degree $\neq 0\Rightarrow f^*:H^k(M)\to H^k(N)$ injective for all $k$

Let $M^n,N^n$ be compact, orientable, smooth manifolds. If $f:N\to M$ is a nonzero degree smooth map, prove that $f^*:H^k(M)\to H^k(N)$ is injective for all $k=0,...,n$. For $k=n$ and $\omega\in\...
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1answer
177 views

If $M$ is a compact $d$-dimensional manifold with nonempty boundary, then $H^d(M)=0$.

The accepted answer to this question contains the following statement: If $M$ is a compact $d$-dimensional manifold with nonempty boundary, then $H^d(M)=0$. This does not look obvious to me, so I ...
2
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1answer
122 views

Understanding de Rham's theorem.

I've been trying to understand a proof of the deRham's theorem. The statement can be found here: https://en.wikipedia.org/wiki/De_Rham_cohomology#De_Rham's_theorem I'm not stating the theorem here ...
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1answer
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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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1answer
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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 $...
2
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1answer
685 views

Calculation of de Rham complex for real projective space

I've been studying de Rham cohomology and the use of the Mayer-Vietoris sequence to compute the de Rham complex for the real projective space $\mathbb P^d$. First, I divided $\mathbb P^d$ in two open ...
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1answer
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Minimal prerequisite to understand De Rham Cohomology

To understand the concept of De Rham Cohomology formally, what is the minimum background required? That is, what are the critical concepts? For example, a list might be something like: tangent ...
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1answer
60 views

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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0answers
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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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0answers
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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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0answers
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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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0answers
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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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0answers
78 views

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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0answers
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Explanation of De Rham Cohomology [duplicate]

Can someone provide an explanation of the intuition behind De Rham Cohomology, and what exactly one is trying to achieve? Assuming the audience knows of manifolds, tangent spaces, cotagent spaces, ...
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0answers
51 views

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(...
2
votes
1answer
89 views

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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vote
3answers
360 views

De Rham cohomology of $S^1$

I want to calculate $H^1_{dR}(S^1)$. I'm stuck at the beginning. I know that $$H^1_{dR}(S^1) = Z^1/B^1 = \frac{\{\alpha\in\Omega^1\ \vert\ X_0(\alpha(X_1)) - X_1(\alpha(X_0)) = \alpha([X_0,X_1])\}}{...
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vote
1answer
251 views

So what Cohomology is? [closed]

I have few questions about Cohomology, all related to each other. Please assume I have minimal knowledge of the subject and I need to have even basic things explained. 1) What is Cohomology? On the ...