Tagged Questions

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Supplement for reading Group cohomology from Serre Local Fields

I am doing a reading course on Group cohomology... I am supposed to start reading Group cohomology part in Serre's Local fields Book ...
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Quantum Cohomology of Affine Toric Varieties

I would like to know whether quantum cohomology rings of affine toric varieties have been calculated, if this is possible. Does anyone have a relevant paper they could refer me to? I have seen it ...
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Generalisation of cochain complexes and “curvature”

Someone has mentioned to me that generalizations of co-chain complexes and their cohomology have been studied, where instead of $d^2 = 0$ we have something like $d^2 \alpha = q \alpha$, which is ...
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Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...
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Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
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Intuition for chains and cochains

I'd like to get some "geometric," "physical," (or other form of) intuition for chains, cochains, and their relationship to integration on manifolds at an elementary level. In particular, it would be ...
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Reference request for bounded cohomology

I want to read Gromov's IHES paper Volume and bounded cohomolgy. I have a decent background in algebraic topology at the level of Hatcher. What other background is required to understand the landmark ...
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Exercises - “From calculus to cohomology”

I am reading Madsen's book From calculus to cohomology and I've found it doesn't have any (explicit) exercises at the end of each section. I'd like to know a few books where I can find some problems ...
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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 ...
327 views

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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Long exact sequence for cohomology with compact supports

Related to my previous question here. Let $X$ be a topological space and let $H_c^{\bullet}(X)$ denote its singular cohomology with compact supports (rational coefficients). Let $U$ be an open subset ...
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Chern Character Isomorphism for non-CW complexes

Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \mathrm{H}^\ast(X; \mathbb{Q})$, where $\mathrm{H}^\ast$ denotes ...
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Cohomology of $\Bbb CP^{\infty}=BU_1, BU_2,\dots$ : A reference request

Where can I find the calculation of the cohomology rings of the classifying spaces $BU_n,~BO_n$ and $BO,~BU$? I took a class where extensive use was made of these cohomology rings, but I missed the ...
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“Inverse problem” for Brauer groups

This question is really just a curiosity, but I'm really interested in the answer. Given a field $K$, we can form the set$^*$ $Br(K)$ consisting of equivalence classes of finite-dimensional central ...
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Twisted Cech cohomology

Let $X$ be a CW-complex with contractible universal cover $\tilde{X}$ and fundamental group $\pi = \pi_1X$. Twisted (co)homology is found by lifting the cell structure on $X$ to a $\pi$-invariant ...
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Twisted de Rham Cohomology

Let $M$ be a smooth manifold and $H$ a closed odd-degree form. Then $(\Omega^{\bullet}(M), d_H)$ defines a complex where $d_H := d + H\wedge$. The cohomology of this complex is called twisted de Rham ...
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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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Top cohomology detecting compactness

Could someone point me to a standard reference for the fact that the top cohomology $H^n(M,A)$ of an $n$-dimensional manifold $M$ is non-trivial for local coefficients $A$ if and only if the manifold ...
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Translation Request - Grothendieck's Tohoku Paper

I've been learning sheaf cohomology, and was interested in reading Grothendieck's Tohoku paper. However, I don't read French. I've done a semi-extensive google search, and the majority of links end ...
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Best book to learn local cohomology with a global point of view

Can anyone suggest me some book that we can use for studying local cohomology, with a global view to many different area of maths, i.e to Commutative Algebra, Number Theory, Algebraic Geometry.... I ...
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cohomology fiber bundles

I will be infinitely grateful to the one who could give a thorough introduction with examples on fiber bundles or a link to a document that deals with it. I was desperately looking on the web for ...