1
vote
1answer
21 views

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 ...
1
vote
1answer
32 views

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 ...
3
votes
1answer
26 views

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 ...
2
votes
1answer
42 views

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 ...
6
votes
1answer
95 views

Cohomological Whitehead theorem

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 ...
3
votes
1answer
62 views

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 ...
1
vote
2answers
67 views

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 ...
0
votes
2answers
107 views

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 ...
1
vote
1answer
58 views

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 ...
2
votes
1answer
314 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 ...
6
votes
1answer
436 views

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 ...
1
vote
0answers
59 views

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 ...
4
votes
1answer
68 views

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 ...
8
votes
1answer
126 views

“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 ...
15
votes
1answer
352 views

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 ...
3
votes
0answers
152 views

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 ...
4
votes
0answers
170 views

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 $E$ ...
2
votes
1answer
204 views

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 ...
4
votes
1answer
874 views

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 ...
4
votes
2answers
295 views

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 ...
1
vote
1answer
242 views

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 ...
7
votes
1answer
854 views

References for calculating cohomology rings

I am struggling to calculate homology rings. Even for a simple space such as the sphere, it is easy to calculate the cohomology, but I find it much harder to find the ring structure. (This link gives ...
11
votes
5answers
1k views

Poincare Duality Reference

In Hatcher's "Algebraic Topology" in the Poincare Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's ...