Questions tagged [de-rham-cohomology]

This a cohomology theory for smooth manifolds, where the (co)chain complex is defined by differential forms on a smooth manifold with differential given by exterior derivative. Then $n^{th}$ de Rham cohomology group is the quotient "closed $n$-forms/exact $n$-forms". Use in conjunction with other algebraic topology and differential geometry related tags if necessary.

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39 views

Confusion with exercise 6.43 in Bott Tu

I am currently working through the book "Differential Forms in Algebraic Topology" by Bott and Tu, though I am confused with one of the problems. Problem 6.43 on page 75 asks: "... Find ...
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Generators of $H^1_{dR}(S^1)$ and angular forms

I am trying to understand and "visualize" the generator of the first de Rham cohomology group of the circle and the intuition behind the concept of angular form. $H^1(S^1)$ can be computed ...
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65 views

Cohomology of $G/T$ for complex reductive group $G$

Okay, I will try this again, last time people didn't like how I asked it :) So let's say $G$ is a reductive complex algebraic group; it could be $GL_n(\mathbb{C})$ if that makes you happy (and in ...
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What is the "generalized degree" of a map $f: \tilde B \to \mathbf G$?

Let $\mathbf G$ be a compact, simple, and simply-connected Lie group. Let $G: \Sigma \to \mathbf G$ be a smooth map, where $\Sigma$ is a Riemann surface. Also, let $\tilde G$ be an extension of $G$ to ...
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100 views

Two definitions of a degree of map $f: S^n \to S^n$ equivalent?

Given a smooth map $f: S^n \to S^n$, I have seen at least two ways of defining the degree. Definition 1: $\deg f$ is an integer satisfying, for every $\omega \in \Omega^n(S^n)$, $$\int_{S^n} f^*\omega ...
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de Rham cohomology on $\mathbb R^n$ is equal to $\{0 \}$ for $k>0$, $\mathbb R$ for $k=0$.

Let $\Omega^k (U)$ be a set of $k-$ diffrential form on $U$, where $U \subset \mathbb R^n$ is region, and define $ Z^k (U)$ be a set of $k-$ closed form, $B^k (U)$ be a set of $k-$ exact form. Then, ...
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41 views

Degree of smooth map (between same-dimensional manifolds) expressed in two ways

Let $M,N$ be connected compact oriented smooth manifolds (without boundary), both of dimension $n$. Let $f : M \rightarrow N$ be a smooth map, let $\omega \in \Omega^n(N)$ be a smooth top-degree form. ...
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39 views

Hodge dual of differential forms $\Rightarrow$ de Rham cohomology and singular cohomology

We know that the de Rham cohomology is isomorphic to the singular cohomology, does the Hodge dual of differential forms induce a dual operation on de Rham cohomology, hence also on singular cohomology?...
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Conventions regarding the de Rham complex

Let $M$ be a smooth $m$ dimensional manifold, $\Omega^k(M)$ the $\mathbb R$-vector space of smooth $k$-forms on $M$ and consider the two cochain complexes $$ 0\rightarrow \Omega^0(M)\xrightarrow{\...
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L^1 Gradient bounds for potentials of weakly closed forms.

The Poincare-lemma is a central statement in differential geometry. It shows that a k-form is closed iff it is exact. A special case is as follows: Let $\omega\in\Omega^k(U)$ with $\omega=\sum_{I\in\...
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Lefschetz fixed-point theorem on Bott and Tu

In the page 129 of the book Bott&Tu, they give an exercise that prove the Lefschetz fixed - point theorem. More percisely, for a compact orientable manifold $M$ with dimension $n$, they assume the ...
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Kunneth formula Bott and Tu

I'm reading Bott and Tu's Differential forms in algebraic topology and I'm stuck on their proof of the Kunneth's formula prop 9.12 for when one manifold has finite dimensional de Rham cohomologies. ...
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Top Dolbeault cohomology group vs top De Rham cohomology group of a compact complex manifold

I cannot reconcile two simple facts for a compact complex manifold $X$ with $dim_{\mathbb{C}}=n$. Let $A^{p,q}(X)$ be differential forms of degree $(p,q)$ on $X$. One the one hand, $$H^{2n}_{DR} (X)= ...
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Analogous of Poincaré Duality for relative homology and relative cohomology

I am studying Morse Theory on finite dimensional and compact manifolds using homology groups and relative homology groups on $\mathbb{Z}$. I want to show that this theory could be developed using De ...
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Cohomological dimension of truncated algebraic De Rham complex.

Is true that the cohomological dimension of the (fine) truncated algebraic De Rham complex $\tau^{\leq q}\Omega^{\bullet}$ is at most $2q$? i.e. its sheaf cohomology vanishes above $2q$. If necessary ...
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Counterexample to Kunneth Formula

I'm reading Example 9.14 of Bott&Tu's Differential Forms in Algebraic Topology, which gave a counter example to the Kunneth formula when the assumption is not satisfied by constructing manifolds $...
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question about dolbeault cohomology

I'm the beginner in complex geometry, and I'm curious about the motivation of dolbeault cohomology,I know that group$H^{q,0}(M)$ represents the holomorphic forms of the manifold $M$,but what's about ...
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61 views

What does the sentence 'de rham cohomology is functorial on the category of continuous maps between open sets in Euclidean spaces' mean?

From the book from Calculus to Cohomology, intro remark in Chapter 6 Homotopy. My query is what does the word functorial mean?
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Application of de Rham Theorem on T^2

As an application of the de Rham theorem on $S^n$, it can be determined that all closed $k$-forms are exact because $H^k_{dR}(S^n) = 0$ for $0<k<n$ (referenced from https://scholar.harvard.edu/...
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Question on the proof of de Rham's theorem

In John M. Lee's Introduction to Smooth Manifolds, in chapter 18, he seeks to prove the de Rham theorem. Step 4 of the proof is to show that if $M$ is a smooth manifold with a de Rham basis, then $M$ ...
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Square differential forms in cohomology

Let $X$ be a differentiable manifold (connected, compact, orientiable) of dimension $4n$. Consider on $X$ a closed $2n$-form $\omega$, with associated cohomology class $[\omega] \in H^{2n}(X,\mathbb{R}...
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$\mathcal{O}$-module action on De Rham cohomology class on open Riemann surface

Let $X$ be an open Riemann surface (or more specifically a compact Riemann surface minus one point). Let $\omega$ be a closed holomorphic $1$-form on $X$. For background, a closed Riemann surface $Y$ ...
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133 views

De Rham cohomology of degree $0$ of a manifold with infinitely many connected components

Exercise 24.2 of Tu's An Introduction to Manifolds (2nd ed.) goes as following: Suppose a manifold $M$ has infinitely many components. Compute its de Rham cohomology vector space $H^0(M)$ in degree $...
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Is there an analogous concept for De Rham cohomology in the framework of Clifford algebras?

I’ve recently read about Clifford’s geometric algebra being a more general framework for differential geometry than differential forms, simpler for the study of spaces with a metric tensor, and ...
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Can $\pi$ be defined in a p-adic context?

I am not at all an expert in p-adic analysis, but I was wondering if there is any sensible (or even generally accepted) way to define the number $\pi$ in $\mathbb Q_p$ or $\mathbb C_p$. I think that ...
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Proof of Thom Isomorphism from Bott & Tu

In chapter 2 of Bott & Tu's book, they supposedly prove the Thom isomorphism in the following way. Firstly, by a trivial spectral sequence, they establish the following isomorphism: $$H_{cv}^*(E)\...
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172 views

Hodge star duality and the metric

Let $X$ be a smooth compact Riemannian manifold of even dimension $2n$. Using the Hodge star $*: \Omega^r(X) \to \Omega^{2n-r}(X)$ one can define self-dual and anti-self-dual $n$-forms on $X$, $$ \...
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719 views

Vanishing differential forms in cohomology

Let $X$ be a smooth differentiable manifold. Consider on $X$ a closed $p$-form $\eta$ and a closed $q$-form $\omega$, which have associated cohomology classes $[\eta] \in H^p(X)$ and $[\omega] \in H^q(...
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A different look on Chern-Weil Theory and the Cohomology Class

I'm studying the Chern-Weil Theory in a book that makes a construction to prove that the cohomology class of c(A) is independent of the connection (and yes, I know that there is an alternate proof for ...
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Derived pushforward of Deligne's complex to the Zariski site.

Deligne complex $\mathcal{D}(p)$ on a smooth variety over $\mathbb{C}$ is defined as the brutal truncation (at level $p$) of holomorphic De Rham complex and also includes the constant sheaf $\mathbb{Z}...
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Closed, Non-Exact $1$-Form on $\mathbb{R}^3 \setminus S^1$

I understand how to compute the de Rham cohomology of $X = \mathbb{R}^3 \setminus S^1$. Based on $H^1(X) \cong \mathbb{R}$, there should be a closed $1$-form on $X$ which is not exact. However, I'm ...
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Induced map on cohomology is preimage in geometry

My question is about p. 69 of Bott Tu. I don't understand how the commutative diagram implies that if $\omega$ is the cohomology class on $M$ representing $S$, then $f^*\omega$ represents $f^{-1}(S)$. ...
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61 views

Representing elements of $H^1(X, K_X)$ on a curve

This is a question about exercise 2 (c), pp.113-114, from Chapter 4 of Voisin's "Hodge Theory and Complex Algebraic Geometry, I". Let $X$ be a compact complex curve with a divisor $D = \sum ...
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Lee's Proof on Top Cohomology of Orientable Noncompact, Connected Manifold

My question is about the theorem $17.32, pp. 455-456$ in Lee's book "Introduction to Smooth Manifolds," second edition. With the hypotheses as in the title of the question, the conclusion is ...
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72 views

Cohomology group of some homogeneous spaces obtained from Lie groups

Here we hope to confirm the cohomology group of a manifold that behaves as some homogeneous spaces obtained from Lie groups. In particular with the coefficients of mod 2 or finite order 2 $\mathbb{Z}/...
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Cohomology group of Grassmannian manifold

Here we hope to confirm the cohomology group of Grassmannian manifold, also this manifold behaves as some homogeneous spaces obtained from Lie groups. In particular with the coefficients of mod 2 or ...
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What's wrong with this computation of algebraic de Rham cohomology of a disjoint union?

Let $R$ be a ring, $S = \mathrm{Spec}(R)$. The algebraic de Rham cohomology is $$H^i_{dR}(S/R) \cong \begin{cases} R & i=0 \\ 0 & i>0 \end{cases}$$ Then I would expect of the disjoint union ...
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Interpretation of Integral of Differential 3-Form over 3-Manifold in Engineering?

So, the standard interpretation of a differential 1-form's integral over a compact, connected, oriented 1-manifold is work, and the standard interpretation of a differential 2-form's integral over a ...
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Closed/exactness of covariant derivatives of differential forms

Is there a condition on vector fields $X$ and/or the underlying manifold $(M,g)$ such that $\nabla_X \omega$ is a closed differential form? That is, $d \nabla_X \omega = 0$. Let us assume for now that ...
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How to proof that a manifold covered by two charts satisfying $f_i(\mathbb{R}^m \setminus B^o) \subset f_j(B^o)$ for $i\not=j$ is simply connected?

While studying for my comprehensive exam, I came across the following problem, which I am unable to solve. Let $B$ denote the closed unit ball centered at the origin in Euclidean space $\mathbb{R}^m$, ...
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Where in algebraic topology does category theory become essential?

I recently finished a first graduate course in algebraic topology, and my institution's final undergrad course in smooth manifolds. Since courses ended, I have been studying de Rham cohomology, in the ...
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Explicit example of Integral de Rham cohomology

I want to see an explicit example of integral de Rham cohomology to gain some intuition that how it work really. More precisely (if I am not mistaken) A $k$-form $\omega$ is integral (i.e. $[\omega]\...
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The ring cohomology of the complex Grassmanian $G_k(V)$

I am trying to understand proposition 23.2. of Bott Tu proof. It wants to prove that $H^{*}(G_k(V))=\frac{\mathbb{R}[c(S),c(Q)]}{(c(S)c(Q)=1)}$. It claims that: 1.$H^{*}(F(V))=\frac{\mathbb{R}[x_1,\...
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Show that $P(V^{*})$ may be interpreted as the set of all hyperplanes in $V$ ($V^{*}=\operatorname{Hom}(V,\mathbb{R}))$

I am starting to study cohomology of manifolds using the Bott To book. Trying to solve some exercises in the book I have run into a problem and I little bite lost on it. Let $V$ be a real vector space ...
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Why non closed differential forms do not play important role for the topology of a manifold?

The question may seems very obvious but I don't know the answer right now. I know that De Rham cohomology reveal some properties of the topology of smooth manifolds by finding closed differential $k$-...
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25 views

Poincare duality of the disk in $\mathbb R^3-S^1$

Construct a 1-form $\sigma$ in $\mathbb R^3-S^1$ satisfying: (a) supp($\sigma$) is bounded in $\mathbb R^3$; (b) for arbitrary smooth embedding $\gamma :S^1 \to \mathbb R^3-S^1$, the integral $\int_{\...
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298 views

Is there a name for this constant associated to smooth maps between spheres? (not degree)

Consider the following constant associated to smooth maps $F: S^{2n-1} \to S^n$ for $n \geq 2$: Let $\omega \in \Omega^n(S^n)$ be a volume form with $\int_{S^n} \omega = 1$. Then there exists $\eta \...
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Computing cohomology over a presheaf

I am reading Bott Tu Differential Forms in Algebraic Topology, and I am having some troubles to do some of its exercises. In chapter 13 (monodromy), exercise 13.7 page 152: The universal covering $\...
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Help proving locally constant presheaf is constant

I am reading Raoul Bott and Loring W. Tu Differential Forms in Algebraic Topology, and in the page 146 there is the next theorem: Let $\mathfrak{U}$ be a good cover on a connected topological space $...
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17 views

Curl of cross product of two vector-valued functions

In Madsen and Tornehave's From Calculus to Cohomology Page 4, there's a formula which the text says can be obtained by straightforward calculations but I don't know how, I hoped to use vector calculus ...

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