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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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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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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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 $\eta \... 1answer 117 views Computing cohomology over a presheaf I am reading Bott Tu Differential Forms in Algebraic Topology, and I am having some troubles to do some of its exercises. In chapter 13 (monodromy), exercise 13.7 page 152: The universal covering$\...
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 \$...