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

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2
votes
1answer
55 views

show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$

I am reading John Lee's book and there is a problem on De Rham cohomology: show that the compactly supported De Rham cohomology groups $H^{p}_{DR}(\mathbb{R^n})$ are all zero for $0\leq p<n$. My ...
1
vote
1answer
34 views

Vanishing of local cohomology groups

Let $k$ be a field and let $X$ be a smooth separated $k$-variety. Let $T$ be a closed integral subscheme of $X$ of generic point $\eta$. The object of interest here is the local cohomology group $$ ...
3
votes
1answer
125 views

Universal coefficient theorem with ring coefficients

The universal coefficient theorem for cohomology reads: $$0 \to Ext(H_{n-1}(C), R) \to H^n(C;R) \to Hom(H_n(C), R) \to 0,$$ where $C$ is a chain complex of free abelian groups and $R$ is a ring. It ...
2
votes
1answer
34 views

Is there any simple example that $lim^1$ terms appear?

limit of cohomology does not behave well in the sense that there will be $lim^1$ term. Is there any simple example that $lim^1$ terms appear? Thanks!
8
votes
1answer
257 views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
4
votes
1answer
237 views

Vanishing of a local cohomology module

I guess $$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$ It is well known $\operatorname{Supp} H^i_I(M)‎\subseteq V(I)\cap \operatorname{Supp}(M)$, therefore $$\operatorname{Supp} ...
1
vote
3answers
72 views

Cohomology groups of real projective space

My question concerns the cohomology groups $H^k(RP^n,\mathbb{Z}_2)$. We know that $H_k(RP^n,\mathbb{Z}_2) = \mathbb{Z}_2$ if $0 \leq k \leq n$ and is trivial otherwise. I looked up the solution and it ...
10
votes
3answers
835 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 ...
9
votes
2answers
502 views

Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
6
votes
1answer
57 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
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 ...
1
vote
1answer
41 views

Homology/Cohomology of Closed Manfold with $\mathbb{Z}_{2}$ Coefficients

Why is $H_{i}(M,\mathbb{Z}_{2}) = H^{i}(M,\mathbb{Z}_{2})$ for a closed manifold $M$? (Hatcher states this on p. 249 in his proof of Corollary 3.37.) Thanks.
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 ...
5
votes
1answer
161 views

Simple exercise in cohomology

I know this is a simple exercise but I am stuck unfortunately. Question: Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
0
votes
1answer
35 views

$\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$

I need help with this problem Let $\sum X =C^+X \cup_X C^-X$ be the union of two cones on $X$ with a common base. Show that $\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$. Dose this give an ...
0
votes
0answers
11 views

Functional isomorphism between $Hom_{\mathbb{Z}}(C,A)$ and $Hom_{\Lambda}(C \otimes \Lambda, A)$

In the proof of the following lemma: Let $G$ be a finite group, and let $A$ be an injective $\mathbb{Z}[G]-module$. Then $A$ is $\mathbb{Z}-injective$, i.e. divisible. It defines $\Lambda=Z[G]$, and ...
2
votes
1answer
85 views

Understanding the Bockstein homomorphism in group cohomology

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism ...
1
vote
1answer
41 views

Natural map of extension groups

Let $\Lambda$ be a cocommutative Hopf algebra over a commutative ring $R$. For two left $\Lambda$-modules $M$ and $N$, interpret $\mathrm{Ext}_{\Lambda}^n(M,N)$ as the set of equivalence classes of ...
1
vote
1answer
95 views

Definition of cohomology with compact support

We can define a cohomology on open manifold: Define a simplicial cochain group $$ \Delta^i_c(X;G)$$ consisting of cochains that are compactly supported in the sense that they take nonzero values on ...
1
vote
0answers
29 views

What is explicit isomorphism between $H^2(G,\mathbb{Q}/\mathbb{Z})$ and $H_2(G,\mathbb{Z})$?

Let $G$ be a finite group. Then its Schur multiplier is the second cohomology group $H^2(G,\mathbb{Q}/\mathbb{Z})$, which is isomorphic to the second homology group $H_2(G,\mathbb{Z})$ (Proof can be ...
2
votes
1answer
133 views

Definition/existence/uniqueness of a minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm trying to understand the following discussion on page $32$ in which he introduces the notion of a minimal projective ...
5
votes
1answer
74 views

Cohomology of the structure sheaf of $\mathbb{P}^1 \times \mathbb{P}^1$

I need to compute the euler characteristic $\chi(\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1})$ of the structure sheaf of $\mathbb{P}^1 \times \mathbb{P}^1$ (the product of the projective complex ...
2
votes
1answer
69 views

equivariant cohomology in case of free actions (basic question)

Suppose $X$ is a topological space and $G$ is a topological group, and $G$ acts on $X$. Here is my question: If $G$ acts freely on $X$, then what are the maps showing $(X \times EG)/G$ is homotopy ...
3
votes
1answer
27 views

Identity in cohomology

Let $N^{4k+1}$ be a compact oriented manifold with boundary $i:M^{4k} \hookrightarrow N$. Suppose $c \in H^{4k}(N,A)$ for some abelian group $A$. I have to prove that $ \langle i^*(c), [M] \rangle =0 ...
1
vote
0answers
30 views

Computing $H^k(\mathbb{C}P^n \times \mathbb{C}P^m, \mathcal{O}^*(\mathbb{C}P^n \times \mathbb{C}P^m))$.

I have to calculate Cech cohomology groups $H^k(\mathbb{C}P^n \times \mathbb{C}P^m, \mathcal{O}^*(\mathbb{C}P^n \times \mathbb{C}P^m))$, working in Cech cohomology, where $\mathcal{O}^*$ is the sheaf ...
8
votes
1answer
76 views

Equality of rank for homology and cohomology groups via the universal coefficient theorem

I'm having trouble understanding a passage from the proof of Corollary 3.37 in Hatcher's Algebraic Topology, namely the fact that the universal coefficient theorem implies $$ ...
0
votes
2answers
47 views

Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
2
votes
1answer
40 views

Stiefel-Whitney Classes: Simple Example

I need help finding the Stiefel-Whitney classes $w_k(\eta)$ of the normal bundle of the $n$-sphere. Now since $H^k (S^n ; \mathbb{Z}/2\mathbb{Z}) =0$ for $k \neq 0,n$, then $w_k(\eta) =0$ for $k ...
0
votes
0answers
11 views

Proof: The dual of the Homology $(H_{n-k})^{*}$= Homology $H_{n-k}$ over the reals?

Proof: The dual of the Homology $(H_{n-k})^{*}$= Homology $H_{n-k}$ over the reals ? So by dual, I mean the linear maps on $H_{k}$. I need this to understand the Poincare duality i.e. $H_{k}\cong ...
1
vote
1answer
42 views

Existence of a suitable cover for $S^{2}$ and a given sheaf

I am trying to find a Leray covering for the 2-sphere with respect to the sheaf $\mathcal{F}=\mathbb{Z}$. I am also assuming that a contractible open covering satisfies $H^{i}(U,\mathbb{Z})$ for all ...
3
votes
1answer
77 views

Question about Relative Cohomology

I need help with the following question please: Suppose that a space $X \subseteq Y $ retracts onto some subspace $A \subseteq X $. When do I have $H^\ast ( Y,X) \cong H^\ast (Y,A)$? Thanks.
0
votes
0answers
32 views

Cohomology of Circle from unreduced Eilenberg-Steenrod Axioms

I would like to compute the cohomology groups of $S^1$ straight from the unreduced Eilenberg-Steenrod axioms. My motivation is to be able to calculate the cohomology group of spheres in any dimension ...
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 ...
2
votes
1answer
57 views

No symplectic structure on $S^{2n},\ n>1$

I am trying to show that there is no symplectic structure on the $2n$-dimensional sphere $S^{2n}$, where $n>1$. I've tried following these steps: (a) Given a compact $2n$-dimensional symplectic ...
2
votes
1answer
115 views

Solving an exercise in Milnor-stasheff's “characteristic classes”

I am trying to solve the following exercise (which is an exercise in Milnor-Stasheff's book). It basically says the following: Let $ M =S^n $ be the $n$-sphere and let $TM$ be its tangent ...
6
votes
1answer
83 views

De Rham cohomology of $T^*\mathbb{CP}^n$

I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me. I want to find the cohomology of $T^*\mathbb{CP}^n$ (seen as a real manifold). Now, this should be equal to ...
2
votes
1answer
53 views

$S^{1}$-bundles over $\mathbb{RP}^2$

How many $S^1$-bundles over $\mathbb{RP}^2$ do exist? Is it true that there exist only two bundles - trivial and not?
1
vote
1answer
46 views

Can we compare cohomology rings with different coefficients?

I have an example sheet that asks me to compute the cohomology rings for two spaces, say X and Y, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_d$ respectively. It then asks whether X and Y are ...
3
votes
1answer
60 views

Relating Ext groups of abelian groups and group cohomology

One can define $\mathrm{Ext}$-groups in the category of abelian groups (not $\mathbb{Z}[G]$-modules) and group cohomology in very similar ways. The second, group cohomology, can be computed in the ...
1
vote
1answer
27 views

Show that $H^{\prime} \cap A$ is a homomorphic image of $M(G)$

Let $H$ be a group and $A$ be a central subgroup of $H$ of finite index. Let $G =H/A$. Show that $H^{\prime} \cap A$ is a homomorphic image of $M(G)$. Here $H^{\prime}$ denotes the commutator ...
5
votes
1answer
95 views

How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a ...
0
votes
1answer
61 views

Dolbeault cohomology on torus

Let $T=\mathbb{C}/\Gamma$ where $\Gamma$ is a lattice of $\mathbb C$. Given that $H_{dR}^1(T)=\mathbb{C}^2$. Prove that $H^{1,0}_\bar{\partial}(T)=\mathbb{C}$. I have no idea what to do. Can someone ...
4
votes
1answer
83 views

If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.

This is an example from a question sheet (non-assessed) of a university class. If M is a non-orientable, closed, connected 3 manifold, prove $H_1(M;\mathbb{Z})$ is an infinite group. I know that since ...
1
vote
0answers
21 views

second hochschild cohomology and extension

i started learning the theorem that says there is a one to one correspondence between Ext(A,M) and H^2(A,M). however, the proof is not clear. I managed to show that there is a well-define map U from ...
1
vote
1answer
42 views

Why do we have $H_2(X,\mathbb Z)\cong\mathbb Z$ for the quintic threefold $X\subset\mathbb P^4$?

Let us work over $\mathbb C$. In this article by S. Katz, it is stated that for a quintic threefold $X\subset \mathbb P^4$ one has $$H_2(X,\mathbb Z)\cong\mathbb Z.$$ Can anyone help me to see why ...
3
votes
1answer
38 views

Existence of Boundary Homomorphisms for Cohomology

I am just starting to learn the basics of cohomology and am confused about the construction of the cohomology groups. So given a group $G$, the idea is you take a projective resolution of $P_0 = ...
2
votes
1answer
93 views

Cohomology to compute number of holes?

Can one use cohomology to compute the number of holes in a space $E$, where $E=R\times R$, $R$ is a Riemann surface of genus $g$, - i.e., is $\dim(H^n(E))$, and by Künneth's formula, $H^{n}(E) \cong ...
2
votes
0answers
41 views

$H^k(\mathbb{C}P^2 \times \mathbb{C}P^2, \mathcal{O}^*(\mathbb{C}P^2 \times \mathbb{C}P^2))$

I'd like to try to compute Cech cohomology groups $H^k(\mathbb{C}P^2 \times \mathbb{C}P^2, \mathcal{O}^*(\mathbb{C}P^2 \times \mathbb{C}P^2))$, but I don't know how can I do it. In my notes the author ...
1
vote
1answer
37 views

Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
0
votes
0answers
37 views

Explititly evaluating the Poincaré duality

Let $M$ be a closed 2-dimensional manifold (a surface). Assume that I have a more or less explicit expression for a Čech 2-cocyle $h_{ijk} \in H^2(M, G)$. I want to know the expression of $h_{ijk}$ as ...