This tag is for questions relating to cohomology groups and cochain complexes.

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2
votes
1answer
368 views

Cohomology of classifying space of torus

I have come heard that the cohomology of the classifying space of a compact torus $T$ is equal to the symmetric algebra over the dual of its Liealgebra $t^*$, where elements of the $t^*$ are of degree ...
1
vote
1answer
169 views

Torsion elements in H^1 of a complex manifold

If $X$ is a compact complex manifold, the exponential sequence gives an injective map $H^1(X,\mathbb{Z}) \to H^1(X,\mathcal{O}_X)$. I think that this shows that $H^1(X,\mathbb{Z})$ is torsion free. ...
7
votes
2answers
231 views

What is the motivation for defining both homogeneous and inhomogeneous cochains?

In my few months of studying group cohomology, I've seen two "standard" complexes that are introduced: We let $X_r$ be the free $\mathbb{Z}[G]$-module on $G^r$ (so, it has as a $\mathbb{Z}[G]$-basis ...
9
votes
4answers
560 views

What do higher cohomologies mean concretely (in various cohomology theories)?

Superficially I think I understand the definitions of several cohomologies: (1) de Rham cohomology on smooth manifolds (I understand this can be probably extended to algebraic settings, but I haven't ...
3
votes
0answers
83 views

Technical question about Brauer groups and smoothness

For a variety $X$ over a field $k$, we define $Br(X) = H^2_{et}(X,\mathbb{G}_m)$. Suppose $X$ is a smooth variety (finite type, separated) over an algebraically closed field $k$ together with a ...
9
votes
1answer
378 views

Étale cohomology of projective space

I have some very basic question about étale cohomology. Namely I would like to compute the étale cohomology of of the projective space over the algebraic closure of $\mathbb F_q$ along with its ...
10
votes
2answers
389 views

Does every Poisson bracket on a commutative algebra come from a second-order deformation?

Let $A$ be a commutative algebra over a field $k$ (of characteristic not equal to $2$ to be safe). Recall that $f : A \otimes A \to A$ is a Hochschild $2$-cocycle if it satisfies $$f(ab, c) + f(a, b) ...
3
votes
1answer
95 views

cohomology of the additive group of imperfect field

In Springer's Encyclopaedia of Mathematics> Galois Cohomology, it is mentioned that For an imperfect field $k$, $H^1(k,\mathbb{G}_a)\neq 0$ in general. I'm looking for such an example or a ...
2
votes
0answers
367 views

Computing the cohomology ring of $\Sigma_2$

This question is from an old exam. I was completely lost on it and not sure where to start- hoping for even a point in the right direction. Let $\Sigma_2$ be the genus 2 surface and ...
1
vote
1answer
283 views

Intersection of two homology classes

Studying the first pages of Gompf-Stipsicz's 4-Manifolds & Kirby Calculus forced me to worry about the geometric meaning of homology and cohomology classes; in particular page 7 contains the ...
5
votes
1answer
382 views

Hodge Number Jump in Family Example

This is based on a comment here: http://mathoverflow.net/questions/67485/can-proper-smooth-base-change-be-used-to-show-that-varieties-cannot-be-lifted-to I felt funny about the comment and I tried to ...
6
votes
2answers
584 views

Cohomology ring of a product

I am trying to calculate $H^*(\mathbb{R}P^3 \times \mathbb{C}P^5,\mathbb{Z})$ as a cohomology ring. I know that $$H^*(\mathbb{R}P^3,\mathbb{Z}) = \frac{\mathbb{Z}[\alpha,\beta]}{(2 \alpha, ...
10
votes
1answer
344 views

Finite groups with periodic cohomology

I'm trying to understand Chapter 12, Section 11 in Cartan + Eilenberg's Homological Algebra, which concerns finite groups with periodic cohomology. Unfortunately I am jumping right to this section in ...
15
votes
2answers
716 views

Meaning of “efface” in “effaceable functor” and “injective effacement”

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...
3
votes
3answers
474 views

What is the connection between Grothendieck's Differential Operators and Hochschild Cohomology

For a given commutative algebra $A$ over a field $\mathbb{K}$(with char=0) the algebra of differential operators on $A$ is the set of endomorphism $D$ of $A$ such for some $n$ we have that for any ...
0
votes
0answers
158 views

transversal intersection and Poincaré duals

If I have $A$, $B$ two submanifolds of dimension n each included in a $2n-$manifold $M$ whose n-cohomology group is free of rank 1 and generator $\alpha$ .denote $\epsilon_{A}$ and $\epsilon_{B}$ both ...
1
vote
1answer
136 views

cohomology of product

I shall be thankful to you for helping me understand what I have highlighted in yellow. I see that $\gamma, \alpha$ and $\beta$ are not the same as the generators of homologies but rather the ...
1
vote
1answer
68 views

reconcile two different cohomolgies

I am in the process of convincing myself of certain results: I see that in the compact support cohomology mayer vietoris has opposite rows compared to the one in singular topology however it is said ...
1
vote
0answers
176 views

polynomial cohomology

Hope this finds you all well. I want to make sure of one thing : Do we usually have polynomial cohomology only in case the cohomology modules are free of rank 1 at most in each degree? PS:I don't ...
1
vote
0answers
93 views

(co)homology of products

let us suppose that we are computing homology of a product where none of the requirements of künneth theorem are valid : is there a general way to compute the homology of such products? Many thanks
1
vote
1answer
244 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 ...
6
votes
3answers
866 views

What are cohomology rings good for?

I am studying some concepts of algebraic topology myself, and I read lately a bit about cohomology rings (created by the direct sum of cohomology groups) but besides all definitions I could not find ...
3
votes
0answers
202 views

group cohomology with coefficient in an induced module

We say that a $G$-module $I$ is induced if $$I\cong L\otimes\mathbb{Z}G$$ where $L$ is an abelian group and the action on $L\otimes\mathbb{Z}G$ is given by the action of $G$ only on the second ...
2
votes
1answer
480 views

cohomology with compact support

Where is the cohomology with compact support useful? It seems that, a part from proving Poincaré duality, we also use it to compute the top dimensional cohomology group of closed manifolds: isn't ...
15
votes
1answer
547 views

Group cohomology versus deRham cohomology with twisted coefficients

Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated ...
7
votes
1answer
875 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
1answer
274 views

What functor does $K(G, 1)$ represent for nonabelian $G$?

For $G$ an abelian group, the Eilenberg-Maclane space $K(G, n)$ represents singular cohomology $H^n(-; G)$ with coefficients in $G$ on the homotopy category of CW-complexes. If $n > 1$, then $G$ ...
9
votes
3answers
825 views

Cohomology easier to compute (algebraic examples)

There is a previous post about motivating cohomology and it contains much of differential geometry examples, something I have just started and still have to figure out. It is said that one uses ...
4
votes
0answers
125 views

What is a high level reason for the fecundity of (co)homological algebra?

A colleague once disparaged his own research to me by saying that it didn't involve any sort of cohomology. It does, in fact, seem like homological ideas appear across disciplines...and are ...
11
votes
1answer
520 views

vector bundles on affine schemes

Serre's theorem (one of them) states that for a quasi-coherent sheaf $\mathscr F$ on an affine noetherian scheme $H^i(X,\mathscr{F})$ vanish for $i >0$. I used to think that this would imply that ...
6
votes
2answers
683 views

Serre Spectral Sequence and Fundamental Group Action on Homology

I am looking at my algebraic topology notes right now, and I am looking at our definition for the Serre Spectral Sequence and it requires that the action of the fundamental group of the base space of ...
6
votes
2answers
304 views

No torsion in $H^1_c(X,\mathbf{Z})$?

If $X$ is a very nice topological space, for example a finite simplicial complex, then is it true that the cohomology with compact supports $H^1_c(X,\mathbf{Z})$ is torsion-free? I have seen an ...
65
votes
2answers
3k views

Direct proof that the wedge product preserves integral cohomology classes?

Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...
5
votes
2answers
810 views

Cohomology of complex projective plane

How can I compute Cohomology of complex projective plane $CP^2$? Any magic like the one here?
20
votes
1answer
1k views

Cohomology of projective plane

How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
2
votes
3answers
449 views

Acyclic vs Exact

I have a question about the words "acyclic" and "exact." Why does Brown use the term "acyclic" instead of "exact" in his book Cohomology of Groups? It seems to me that these two terms exactly ...
3
votes
2answers
461 views

cup product well definedness

So the cup product is not well defined over co-chain groups, but all the books claim it is well defined over co-homology groups. The only thing I am not clear on is invariance under ...
7
votes
2answers
289 views

Manifold with 3 nondegenerate critical points

Suppose $M$ is a n-dimensional (compact) manifold and $f$ is a differentiable function with exactly three (non-degenerate) critical points. Then one can show, using Morse theory, that $M$ is ...
3
votes
1answer
109 views

Pushing forward sheaves and the result on sheaf cohomology

Let $f:X \rightarrow Y$ be a continuous map of topological spaces and let $\cal{F}$ be a sheaf on $X$. Is there an obvious map $H^\ast(Y,f_\ast \mathcal{F} ) \rightarrow H^\ast (X,\cal{F})$?
15
votes
3answers
1k views

How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
8
votes
1answer
599 views

Homology of the Empty set

I am under the impression that the standard convention for the homology (singular) of the empty set is 0 in all non negative degrees and $\mathbb{Z}$ in degree $-1$. I have no problem with this ...
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 ...
16
votes
3answers
1k views

Motivating Cohomology

Question: Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology? Why ...
6
votes
2answers
2k views

Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...
4
votes
1answer
267 views

Cohomology of $\mathcal O_X$ for toric varieties

Motivated by my ignorance here, if $X$ is a projective toric variety, is $$H^m(X, \mathcal O_X) \cong \begin{cases} 0 & m > 0 \\ \mathbb C & m = 1 \end{cases} $$ as for $\mathbb ...
13
votes
1answer
716 views

Topological vs. Algebraic $K$-Theory

Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras ...
4
votes
1answer
282 views

Quotient spaces and equivariant cohomology

Consider a $G$-equivariant map $\pi:X\to Y$ for $G$ an affine algebraic group, such that $\pi$ is a good categorical quotient. Is there any relationship between $H^*_G(X)$ and $H^*(Y)$? Is there if ...
2
votes
1answer
91 views

connecting maps in the universal coefficients theorem

for a finite group G and a trivial abelian G module A, there is the short exact sequence $0 \rightarrow Ext^1 (G_{ab},A) \rightarrow H^2(G,A) \rightarrow Hom (H_2 (G,Z), A) \rightarrow 0$ I'm ...
6
votes
1answer
251 views

Is there any relation about rational homology of X and X/G

If we know the rational homology of X is 0, can we get some information about the rational homology of X/G, where G is a finite group? Thank you very much for the answers!
2
votes
1answer
458 views

Cohomology of line bundles on the blowup of $\mathbb P^2$

Let $X$ denote the blowup of $\mathbb P^2$, $E$ the exceptional divisor, and $H$ the pullback of the hyperplane class. How can I compute $H^0(X,mE+nH)$, $H^1(X,mE+nH)$, and $H^2(X,mE+nH)$ for $m,n ...