# Tagged Questions

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### De Rham cohomology of the pointed plane

i try to work out some examples for de Rham cohomology, but i have some problems: I want to figure out what $H^k(\mathbb{R}^2\setminus\{0\})$ is and want to generalize this to arbitrary finite points ...
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### Chains and cochains: integer versus real coefficients

Let a real, smooth manifold $M$ be given. For each non-negative integer $k$, let a singular $k$-cube on $M$ be a continuous mapping $c:[0,1]\to M$. Let $C_k(M,\mathbb Z)$ denote the set of formal ...
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### Exact and closed forms with a vanishing Riemann tensor

I need a result to prove that an closed form is equally exact. I work under the assumption that the Riemann tensor vanishes everywhere on the manifold. (It is the context of general relativity.) The ...
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### Top de Rham cohomology

I just realized that I never really understood why $H_{dR}^n(M, \mathbb{R}) = \mathbb{R}$ if $M$ is compact and $H_{dR}^n(M, \mathbb{R}) = \{0\}$ if $M$ is not compact (provided that's true?). I'm ...
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### how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
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### de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
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### Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
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### Differential forms as functionals on curves

Please give me a reference to a book or lecture notes where the following stuff is studied. Let $M$ be a Riemann surface with boundary $\partial M$ (but not necessarily, any smooth $n$-dimensional ...
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### Definition of the relative de-Rham cohomology and its generalization

Let $M$ be a smooth manifold and $N$ be its smooth submanifold. We say that two closed forms $\omega_1$ and $\omega_2 \in \Lambda^k(M)$ are equivalent if their difference is an exact form from ...
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### Relative de Rahm cohomology computation for two disjoint circles embedded in R^2

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
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### First proof of Poincaré Lemma

I know that a way of proving Poincare lemma is to use the homotopy invariance and contractibility of the Euclidean space. Is there is a way of doing it directly (without using the contractibility of ...
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### The Poincare Lemma for Compactly Supported Cohomology

IÂ´m reading the proof of The Poincare Lemma for Compactly Supported Cohomology there is a part in the proof that said in the text book Bott and Tu: $d \pi_{\ast} = \pi_{\ast} d$ in other words, ...
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### The degree of every smooth map $\mathbb{R}^n \to \mathbb{R}^n$ is one…

Let $\varphi : M^n \to N^n$ be a proper smooth map between two connected smooth manifolds. Then $\varphi$ induces a linear map $\varphi^* : H_c^n(N) \simeq \mathbb{R} \to H_c^n(M) \simeq \mathbb{R}$ ...
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### De Rham Cohomology of $M \times \mathbb{S}^1$

Let $M$ be a closed (compact, without boundary) $m$-dimensional manifold. I want to prove that $H^{k+1}(M \times \mathbb{S}^1) = H^k(M) \oplus H^{k+1}(M)$. ($H^k$ is the $k$-th De Rham cohomology ...
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### Request for companion of Mariano Suárez-Alvarez's proof.

Mariano SuĂˇrez-Alvarez's answer to Cohomology of projective plane seems very interesting. However, there are three pieces I could not stitch up for one of his proofs. Wonder if someone may help? ...
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### A reference about Dolbeault cohomology

I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.
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### De Rham cohomology notation

According to http://en.wikipedia.org/wiki/De_Rham_cohomology, one defines the $k$-th de Rham cohomology group $H^{k}_{\mathrm{dR}}(M)$ to be the set of equivalence classes, that is, the set ...
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### How do I know when a form represents an integral cohomology class?

Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class. I would like to ...
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### Restriction map on a compact orientable manifold without a boundary.

I have the following problem: Let $M$ be and $n$-dimensional compact oriented manifold without boundary. Let $p\in M$ be and point and let $M_p=M\backslash\{p\}$. Let $j:S^{n-1}\to M_p$ be the ...
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### Cohomologies of $\mathbb R^n$ with rational differential forms

We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
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### Simple exercise in cohomology

I know this is a simple exercise but I am stuck unfortunately. Question: Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
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### De Rham Cohomology of Product of Manifold with an Open Interval

Let $X$ be a submanifold of $\mathbb{R}.$ Prove that $H^{k}_{DR} (X) = H^{k}_{DR} (X\times (0,1)).$ I know that we should consider maps $\iota_a: X\to X\times (0,1)$ by $\iota_a(x) = (x,a)$ for ...
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### Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
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### Induced sequence of global sections

I'm reading Differential Analysis on Complex Manifolds by Raymond O. Wells. It states the following in the beginning of section 3 of chapter 2 on page 51: Consider a short exact sequence of sheaves: ...
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### De Rham cohomology of $\mathbb R^3$ without lines and a circumference

I am trying to calculate De Rham cohomology of the following spaces: $X=\mathbb R^3\setminus r$ where $r$ is a line; $Y=\mathbb R^3\setminus (r \cup \gamma)$ where $r$ is a line and $\gamma$ is a ...
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### What is the solution to Nash's problem presented in “A Beautiful Mind”?

I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
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### When and *why* should one view the space of forms as one big space?

I wonder why in books about differential geometry there seem to be no relevant results about properties of mixed differential forms like \omega_{12}\ \mathbb{d}x^1\land\mathbb{d}x^2+\lambda_{123}\ ...
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### group action - compact complex torus with $H^2_{DR}(X,C) = 0$ (de Rham cohomology)

If $\mathbb{Z}$ acts on $\mathbb{C}^n \backslash \{0\}$ by $(m,z) \mapsto 2^m\,z$, I need to show that $H^2_{DR}(X=(\mathbb{C}^n \backslash \{0\})/\mathbb{Z},\mathbb{C})=0$. I start with showing it ...
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### de Rham Cohomology of Non-Flat Bundle

Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$. If $E$ ...
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### Can a Non-Compact Manifold have Infinite Dimensional Cohomology?

For compact manifolds, Hodge Theory tells us that (de Rham) cohomology is finite dimensional. What about non-compact manifolds? That is: Can non-compact manifolds have infinite dimensional ...
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### ellipticity of the Laplacian associated to the de Rham complex

I am struggeling with the following comment that I read regarding the de Rham complex: Define $(d + \delta)_e : C^\infty(\Lambda^e(T^*M)) + C^\infty(\Lambda^o(T^*M))$ where ...
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### De Rham cohomology for non-compact manifolds

Let $M$ be a non-compact differential manifold. It is true that in general $H^q_c(M) \neq H^q(M)$, where $H^q_c$ is the de Rham's cohomolgy with compact support group and $H^q$ is the usual de Rham's ...
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### Intersection of two homology classes

Studying the first pages of Gompf-Stipsicz's 4-Manifolds & Kirby Calculus forced me to worry about the geometric meaning of homology and cohomology classes; in particular page 7 contains the ...
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### Group cohomology versus deRham cohomology with twisted coefficients

Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated ...
Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...