A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
Questions tagged [cohomology]
1355 questions
68
votes
7 answers
Can anyone give me a good example of two interestingly different ordinary cohomology theories?
An answer to the following question would clarify my understanding of what a cohomology theory is. I know it's something that satisfies the Eilenberg-Steenrod axioms, and I know that those axioms allow you to work out quite a lot. But what sort of…
gowers
- 28,729
39
votes
7 answers
What is a cup-product in group cohomology, and how does it relate to other branches of mathematics?
I have a few elementary questions about cup-products.
Can one develop them in an axiomatic approach as in group cohomology itself, and give an existence and uniqueness theorem that includes an explicitly computable map on cochains? Second, how do…
Akhil Mathew
- 25,291
15
votes
1 answer
The cohomology of a product of sheaves and a plea.
The question Consider a topological space $X$ and a family of sheaves (of abelian groups, say) $\; \mathcal F_i \;(i\in I)$ on $X$. Is it true that
$$H^*(X,\prod \limits_{i \in I} \mathcal F_i)=\prod \limits_{i \in I} H^*(X,\mathcal F_i) …
Georges Elencwajg
- 46,833
9
votes
1 answer
Should cohomology of $\mathbb{C} P^\infty$ be a polynomial ring or a power series ring?
Some people define total cohomology of a space $X$ to be $\bigoplus_{i \geq 0} H^i(X)$, which would make $H^*(\mathbb{C} P^\infty)$ a polynomial ring in one generator of degree 2.
However, it seems like thinking of $H^*(\mathbb{C} P^\infty)$ as a…
8
votes
4 answers
Cohomological dimension
Cohomological dimension arises in many things I read, but my familiarity with it is superficial. What's a good source for understanding cohomological dimension on its various examples, fascets, theorems, and applications, basically providing the…
Makhalan Duff
- 5,819
6
votes
1 answer
determinant of periods
Let $X$ be a projective, smooth algebraic variety defined over the field of algebraic numbers. Consider algebraic de Rham cohomology $H_{dR}(X)$ and singular cohomology of $X(\mathbb{C})$ with rational coefficients $H_B(X)$. One can look at the…
reference
- 71
2
votes
0 answers
Cohomology of a projective limit
If we consider an oriented locally finite graphs $X$, and we suppose that $X$ is the projective limit of a familly of oriented locally finite graphs : $X=\lim\limits_{\overleftarrow{k\in\mathbb{N}}}X_{k}$, Is it true that $$\displaystyle…
Rajkarov
- 933
2
votes
0 answers
Weil's proof of de Rham theorem
de Rham's theorem states the integral of forms on chains induce an isomorphism, while the exact sequence proof not give the isomorphism explicitly. I have see some tedious explanation the two canonical isomorphisms coincide. Are there some general…
MiGang
- 153
2
votes
1 answer
Cohomology of quotient space
Let be $X$ a maximal torus in a Lie group $G$. I'd like to calculate the cohomology $H^{*}(G/N(T))$. I know that it is trivial in odd degree and the base-field is even but I haven't a basic method to do this. Is there a simple method of calculation…
Oscar1778
- 21
1
vote
1 answer
Proof that higher cech cohomology groups vanish for fine sheaves.
So I've been trying to understand a proof of the fact that $H^p(M, \mathcal F)=0$ whenever $p\geq 1$ from page 42 of the book "Principles of Algebraic Geometry" by Griffiths and Harris.
The proof is carried out for the sheaf ${\mathcal a}^{r,s}$ of…
1
vote
2 answers
Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?
Does anyone have an example of two spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?
is it possible?
rhl
- 133
1
vote
1 answer
Is there a classification relating split extensions $G$ by $K$ and homomorphisms $G \to \mathrm{Aut}(K)$?
Are the equivalent class of split extension of $G$ by $K$ really in one to one correspondence with homomorphisms $G \to \mathrm{Aut}(K)$?
When I am trying to prove it, I find it may be not the case.
I only know that
$$1\to K\to K\rtimes_{\rho_1}G\to…
Xuexing Lu
- 747
- 5
- 18
1
vote
1 answer
Projection onto cocycles
Consider a finite simplicial complex $X$ and the simplicial cochain complex with real coefficients. The cochain groups are finite-dimensional vector spaces, they have a natural scalar product. The $n$-cocycles $Z^n$ are a linear subspace in the…
user2412
1
vote
1 answer
first cohomology of exterior power of tangent bundle
suppose G is a Grassmannian manifold, and TG is the tangent bundle.
By Bott's theorem $H^1(G, T_G)=0$.
Is it true that $H^1(G, \bigwedge^i T_G)=$, for i>0.
I saw some vanishing result by lepotier, but would like to confirm.
thanks.
john
- 457
0
votes
1 answer
Crossed module classification theorem and $H^3(G; A)$
We are interested in "Crossed Module Classification Theorem (Two crossed modules with kernel $A$ and cokernel $G$ determined the same class in $H^3(G;A)$ if and only if they are equivalent.)".
We study the works of Charles A. Weibel "An Introduction…
