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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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199 views

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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1answer
747 views

Homotopy invariance of de Rham cohomology

Let $M,N$ be smooth manifolds which are homotopy equivalent i.e., there exists smooth maps $F:M\rightarrow N$ and $G:N\rightarrow M$ such that $F\circ G$ is homotopic to identity map on $N$ and $G\...
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vote
1answer
249 views

So what Cohomology is? [closed]

I have few questions about Cohomology, all related to each other. Please assume I have minimal knowledge of the subject and I need to have even basic things explained. 1) What is Cohomology? On the ...
4
votes
1answer
167 views

How does one introduce characteristic classes

How do you introduce or how are you introduced to characteristic classes. I am assuming the student is comfortable with principal bundles and connections on principal bundles.
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1answer
89 views

Integrating $[\alpha] \in H^1_{dR}$ along a closed curve is well-defined

Let $U := \mathbb R^2 \setminus \{0\}$ be the punctured plane and $\gamma$ the counter clockwise once traversed unit circle. Consider $$\tag{1} [α] \mapsto \int^{2 \pi}_{0} \alpha_{\gamma (t)}(\gamma'...
2
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1answer
684 views

Calculation of de Rham complex for real projective space

I've been studying de Rham cohomology and the use of the Mayer-Vietoris sequence to compute the de Rham complex for the real projective space $\mathbb P^d$. First, I divided $\mathbb P^d$ in two open ...
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1answer
58 views

Homotopy Theory involved in De Rham Cohomology computation

I am trying to understand the computation of a De Rham Cohomology of a Real Projective Space. When we cover $ℙ^d$ with the sets $U=\{[x^0,\dots,x^d]: x^d≠0\}$ and $V=\mathbb{P}^d\smallsetminus \{[0,......
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1answer
43 views

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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1answer
32 views

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}^...