This tag is for questions relating to cohomology groups and cochain complexes.
2
votes
1answer
133 views
Approaching a Cohomology Computation
If $U \subseteq \mathbb{R}^2$ is the complement of $d > 0$ points in the plane, and $H^k$ denotes the $k$-th cohomology group, how should I verify that $H^k (U)$ equals $\mathbb{R}$, ...
11
votes
1answer
331 views
Why isn't $H^*(\mathbb{R}P^\infty,\mathbb{F}_2)\cong \mathbb{F}_2[[x]]$?
We just computed in class a few days ago that $$H^*(\mathbb{R}P^n,\mathbb{F}_2)\cong\mathbb{F}_2[x]/(x^{n+1}),$$ and it was mentioned that $H^*(\mathbb{R}P^\infty,\mathbb{F}_2)\cong \mathbb{F}_2[x]$, ...
8
votes
3answers
222 views
Turning cobordism into a cohomology theory
I've recently finished one semester in differential topology (with Milnor's Topology from the Differentiable Viewpoint) and my first semester of algebraic topology. I believe I understand Milnor's ...
7
votes
4answers
608 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 ...
6
votes
2answers
238 views
Cohomology ring $H^*(\mathbb{R}P^3 \# \mathbb{R}P^3; \mathbb{Z}_2)$.
I am interested in computing the cohomology ring $H^*(\mathbb{R}P^3 \# \mathbb{R}P^3; \mathbb{Z}_2)$. Here # is the connected sum. Using a suggestion here on my earlier post, I computed the additive ...
2
votes
1answer
279 views
computing cohomology algebra of connected sum of two real projective spaces
Could someone tell me what is the cohomology algebra $H^*(\mathbb{R}P^n \# \mathbb{R}P^n; \mathbb{Z}_2)$ and how to compute it. Here $\#$ is the connected sum.
Thanks.
10
votes
2answers
370 views
Which cohomology theories have a formula $\langle \Omega,\text d \omega \rangle = \langle \partial \Omega,\omega \rangle$?
Is a formula
$$\langle \Omega,\text d \omega \rangle = \langle \partial \Omega,\omega \rangle$$
like Stokes theorem
$$\int_\Omega \text d \omega=\int_{\partial\Omega} \omega$$
common in cohomology ...
2
votes
1answer
123 views
computing cohomology algebra of 3 dimensional Klein bottle
Could someone tell me how to compute the cohomology algebra $H^*(K, \mathbb{Z}_2)$ of the three dimensional Klein bottle $K$ defined as follows. Let $S_0,S_1$ be the boundaries of $S^2 \times [0,1]$ ...
4
votes
2answers
1k 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 ...
3
votes
2answers
192 views
What restricts the number of cohomologies?
Do different cohomology theories essentially just exist because there are distinguished homology theories associated with them?
If yes, is it known if there is always a relation like the Poincaré ...
8
votes
1answer
225 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 ...
4
votes
0answers
275 views
De Rham cohomology for non-compact manifolds
Let $M$ be a non-compact differential manifold. It is true that in general $H^q_c(M) \neq H^q(M)$, where $H^q_c$ is the de Rham's cohomolgy with compact support group and $H^q$ is the usual de Rham's ...
4
votes
1answer
72 views
Existence of acyclic coverings for a given sheaf
Let $\mathcal{F}$ be a sheaf over $X$ and $\mathcal{U}=\{U_i\}_{i\in I}$ a covering of $X$.
I say that $\mathcal{U}$ is acyclic for $\mathcal{F}$ if $H^k(U_{i_0 \ldots U_n}, \mathcal{F}|_{U_{i_0 ...
2
votes
0answers
24 views
How to prove $H^2(\mathfrak{g}, J(\mathfrak{g}))\neq0$, where $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$?
$\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over a field $k$ with $\mathrm{char}k=0$. $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$. That is, the kernel of ...
3
votes
1answer
84 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 ...
1
vote
1answer
178 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 ...
9
votes
1answer
275 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 ...
11
votes
1answer
232 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 $G$ ...
5
votes
1answer
133 views
Geometric invariants of a scheme
Following my previous question about sheaf cohomology, I'd like to ask about its applications to algebraic geometry. I have now learned a little about homological algebra and I can see that for ...
4
votes
2answers
106 views
$H^1(G,A)$ is killed by $|G|$ : Proof on the level of cocycles
Let $G$ be a finite group and $A$ a $G$-Module. It is well-known that $H^q(G,A)$ is killed by $|G|$ for all $q \geq 1$. This is usually proved using Restriction-Corestriction (applied with the trivial ...
3
votes
0answers
66 views
Examples showing the usefulness of derived categories
What are examples that show that derived categories really makes things easier/more transparent/have a real use?
8
votes
3answers
615 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 ...
6
votes
1answer
210 views
What tells rational cohomology about integral cohomology?
Say we have a finite CW complex with cells only in even degrees. For example a $\mathbb {CP}^n$ or a complex flag variety. If we know the rational cohomology ring, does it also determine the integral ...
1
vote
1answer
104 views
Group cohomology: $H^1$ of $\mathbb Z/4 \mathbb Z$
Consider the group $G = \mathbb{Z}/4\mathbb{Z}$ and its subgroup $H = 2\mathbb{Z}/4\mathbb{Z}$. Consider the obvious injection $\mathbb{Z} \to \mathbb{Z}[G]: 1 \mapsto N_G = \sum_{\sigma \in G} ...
3
votes
1answer
137 views
Hartshorne Exercise III.2.1(a)
Show that $H^1(\mathbf{A}^1_k, \mathbf{Z}_U) \neq 0$ for $U = \mathbf{A}^1_k \setminus \{P,Q\}$, $k$ infinite field.
Is it really neccessary that $P \neq Q$?
My proof is as follows: Take the long ...
2
votes
2answers
187 views
Is the sheaf of locally constant functions flasque?
Quick question, is the sheaf of locally constant functions flasque?
5
votes
1answer
224 views
Torsion-free virtually-Z is Z
It is well known that a torsion-free group which is virtually free must be free, by works of Serre, Stallings, Swan...
Is there a simple cohomological proof of the fact that a torsion-free group ...
0
votes
0answers
43 views
classifying space of $p$-group
I want to know a model for the classifying space of a finite $p$-group $P$ (say of order $p^n$) and the mod $p$ cohomology algebra of $P$. In particular, what is the classifying space and mod $p$ ...
1
vote
1answer
229 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 ...
4
votes
2answers
162 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 ...
1
vote
1answer
89 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.
...
6
votes
4answers
389 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 ...
4
votes
1answer
213 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 ...
3
votes
0answers
75 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 ...
0
votes
1answer
187 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 ...
2
votes
1answer
314 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 ...
2
votes
0answers
227 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 ...
4
votes
1answer
250 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 ...
4
votes
2answers
321 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, ...
6
votes
2answers
246 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 ...
3
votes
3answers
318 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 ...
10
votes
1answer
584 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 ...
0
votes
0answers
142 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
116 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
66 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
145 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
83 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
4
votes
2answers
529 views
Cohomology of complex projective plane
How can I compute Cohomology of complex projective plane $CP^2$?
Any magic like the one here?
4
votes
3answers
492 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
156 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 ...
