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

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11
votes
1answer
268 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
806 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
489 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
651 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
298 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
802 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
425 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
450 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
281 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
584 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
263 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
697 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
280 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
242 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
451 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 ...
4
votes
1answer
400 views

When does the virtual cohomological dimension become a numerical invariant?

In group cohomology theory, we know that the cohomological dimension cd(G) for a profinite group is a fundamental numerical invariant. We say G is of virtual cohomological dimension n (denote it by ...
10
votes
3answers
824 views

Toy sheaf cohomology computation

I asked this question a while back on MO : http://mathoverflow.net/questions/32689/how-should-a-homotopy-theorist-think-about-sheaf-cohomology One thing that really helped in learning the Serre SS ...
5
votes
1answer
316 views

Varying definitions of cohomology

So I know that given a chain complex we can define the $d$-th cohomology by taking $\ker{d}/\mathrm{im}_{d+1}$. But I don't know how this corresponds to the idea of holes in topological spaces (maybe ...