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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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algebraic de Rham cohomology of normal crossing singularities

For a variety $X/k$ the standard way of defining de Rham cohomology $H^i_{\mathrm{dR}}(X)$ is as the hypercohomology $\mathbb{H}^i(\Omega^{\bullet}_{X/k})$ of the de Rham complex, and this requires $X$...
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Show that every smooth map from $S^n\to T^n$ has degree 0.

Here is my attempt: The degree of a smooth map $f:M \to N$ (where $M,N$ are manifolds of same dimension) is defined on the top form. Since the integral operator induces a natural isomorphism from the ...
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nonzero cohomology class on even dimensional compact manifold

Let $M$ be a compact manifold of dimension $2n$. Let $\omega$ be a 2 form on $M$ such that the induced bundle map $\tilde{\omega}: TM \to T^*M$ defined by $\tilde{\omega}(X)(Y) = \omega(X,Y)$ is a ...
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Proof of the Hairy Ball Theorem

In the proof of the Hairy Ball theorem, we use the diffeomorphisms, $F_t(x)=\cos(\pi t)x+\sin(\pi t)v(x)$ where $v(x)$ is the non-vanishing vector field, then use the fact that $F_1^*=F_0^*$ as ...
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Exterior Product on de rham homology

Given a smooth manifold $M$ and its differential graded commutstive de Rham algebra $(\Omega(M),d,\wedge)$, the wedge product $\wedge$ can be projected onto the de Rham cohomology $(H_{dR}(M),\wedge)$....
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Computing algebraic de Rham cohomology

Let $R=\mathbb C[x,y]/(y(x-a)(x-b)-1)$ where $a,b$ are distinct complex numbers. Show that the cohomology of the de Rham complex $$0\to R\to \Omega_{R/\mathbb C}\to 0$$ is $\mathbb C$ in degree zero ...
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The pullback of $\alpha\in \Omega^k(N)$ by $F:M\times[0,1]\to N$.

Suppose $M$ and $N$ are smooth manifolds of dimensions $m$ and $n$ respectively. Suppose we have a smooth map $F:M\times [0,1]\to N$. Clearly, for $t\in[0,1]$, $F$ defines a smooth map $F_t:M\to N$. ...
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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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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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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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de Rham cohomology of $S^2 \setminus \{$points$\}$

As a follow up to this question, I'm interested in how to calculate the de Rham cohomology groups of $S^2 \setminus \{$n points$\}$, where $n = 2, 4$ and then for general $n$, using Mayer-Vietoris/...
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Homotopy equivalence of $\mathbb{R}^3 \setminus \{$lines$\}$

I'm aware that $\mathbb{R}^3 \setminus \{$a single line$\}$ is homotopy equivalent to $\mathbb{R}^2 \setminus \{$pt$\}$. Similarly, $\mathbb{R}^3 \setminus \{$two parallel lines$\}$ is homotopy ...
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de Rham cohomology of $U(2)$

I'm trying to calculate the de Rham cohomology of $U(2)$, but I don't know how to do this. I'd like to avoid Mayer-Vietoris if possible. I'm doing this in preparation for an exam in my topology course ...
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de Rham cohomology and connected components

I'm trying to work out what $H^0_{dR}(\mathbb{R}^2 \setminus \{p,q\})$ is, and I've come across the fact that the dimension of $H^0_{dR}(M)$ is equal to the number of connected components of the ...
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How should I interpret $H^1(X,\{P_1,\dotsc,P_n\};\mathbb{Z}\oplus i\mathbb{Z})$?

I am reading A. Zorich's article "Square Tiled Surfaces and Teichmuller Volumes of the Moduli Spaces of Abelian Differentials" and there a main object is the space $$H^1(X,\{P_1,\dotsc,P_n\};\mathbb{Z}...
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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\...