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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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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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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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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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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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Cap product and de Rham cohomology

Let $M$ be a compact smooth $d$-dimensional oriented manifold. The natural pairing of $d$-forms $\omega^{(d)}$ with the fundamental class is given by integration $\int_M \omega^{(d)}$. Let us also ...
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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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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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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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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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Apparent contradiction to Poincaré lemma?

I have learned that, if we have a connected, oriented and compact n-dimensional manifold, the top de Rham cohomology is isomorphic to $\mathbb{R}$, i.e $$ H_{\text{dR}}^n(M) \cong \mathbb{R}.$$ ...
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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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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=...