# 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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### 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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### 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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### 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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