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

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2
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0answers
57 views

Long Exact Sequence Cohomology with Compact Support

I found in this topic or (question) a reason for my question, but i do not understand it. As this question is quite old, I hope someone else can help me. Assume $U$ is an open subset of a topological ...
2
votes
0answers
33 views

Cohomology-Homology bilinear form of Seifert surfaces

Let $C_\ast$ be any chain complex of $R$-modules. Then for any $k\in\mathbb{Z}$ we obtain a $R$-bilinear map $$\langle-,-\rangle:H^k\!C_\ast\times H_kC_\ast\longrightarrow R, ...
0
votes
1answer
36 views

On the hypothesis of the Additive Cousin Problem

The Additive Cousin Problem is the following: Assume that $D$ is a region (open, connected) of $\mathbb{C}^n$. Assume that the Dolbeault Cohomology Group of $D$, $H^1_{\bar{\partial}}(D)$ is equal to ...
11
votes
2answers
110 views

Computing cohomology of hypersurface

I'm taking a course on differential geometry now, and we got the following exercise from the lecturer: compute the (de Rham) cohomology groups $H_{dR}^i(M)$ of your favourite space. In all the ...
4
votes
1answer
58 views

Can the cohomology vanish on $X$ but on no restriction to hyperplane sections?

I would need a result asserting that if, say, a locally free sheaf $\mathcal{F}$ on a projective $X$ has non-vanishing cohomology when restricted to any smooth hyperplane section, then $H^1(X, ...
0
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0answers
12 views

How to Resolve Extension Issues in Equivariant (Co)Homology Computations

I am computing equivariant homology, which is just the usual homology of the Borel construction. I have reached a few extension issues. I have used the Gysin sequence, Leray-Serre spectral sequence, ...
4
votes
0answers
80 views

Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
2
votes
1answer
50 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
33 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 $$ ...
2
votes
1answer
98 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
32 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!
1
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3answers
59 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 ...
6
votes
1answer
55 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 ...
1
vote
1answer
34 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.
0
votes
1answer
32 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
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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 ...
1
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1answer
38 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
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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 ...
1
vote
1answer
93 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 ...
2
votes
1answer
75 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 ...
5
votes
1answer
72 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 ...
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 ...
2
votes
1answer
38 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
2answers
46 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 ...
8
votes
1answer
67 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
0answers
10 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 ...
0
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0answers
30 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 ...
2
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1answer
63 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
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1answer
76 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.
3
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1answer
57 views

Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
9
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2answers
478 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 ...
2
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1answer
54 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
101 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
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1answer
82 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 ...
1
vote
1answer
44 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 ...
2
votes
1answer
51 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?
3
votes
1answer
58 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
93 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
59 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 ...
6
votes
1answer
86 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 ...
3
votes
1answer
70 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 ...
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0answers
20 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
41 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
3answers
111 views

Relation between $K$-theories

I apologize in advance if this question is too vague/general. I am curious to know how all of the different $K$-theories are related (algebraic $K$-theory, topological $K$-theory, twisted $K$-theory, ...
3
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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
91 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 ...