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

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25 views

Pro-p-groups and free pro-$p$-groups, an exact sequence (Generators and Relations)

I currently have the following issue (generators and relations for profinite groups): Let $\mathcal{S}={(g_i)}_{i\in I}$ be a system of generators for a pro-$p$-group $G$, then we can obtain an exact ...
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1answer
25 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.
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1answer
18 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 ...
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0answers
9 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 ...
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1answer
26 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 ...
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0answers
24 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 ...
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2answers
50 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 ...
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1answer
51 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 ...
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1answer
57 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 ...
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0answers
27 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
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1answer
19 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 ...
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2answers
40 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 ...
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1answer
44 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 $$ ...
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0answers
8 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 ...
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1answer
33 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 ...
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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 ...
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1answer
50 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 ...
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1answer
72 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.
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22 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 ...
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2answers
406 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
37 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 ...
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1answer
77 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 ...
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1answer
69 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 ...
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1answer
42 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
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1answer
46 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?
2
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1answer
43 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 ...
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1answer
24 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 ...
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1answer
68 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 ...
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1answer
31 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 ...
5
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1answer
56 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
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1answer
48 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
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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 ...
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1answer
35 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 ...
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1answer
42 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, ...
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1answer
36 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
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1answer
71 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
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0answers
38 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 ...
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1answer
32 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 ...
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0answers
32 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 ...
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19 views

worked out examples in borel-moore homology

I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking ...
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0answers
32 views

Algebraic Variety compact cohomology and singular cohomology.

Given an algebraic variety $X$ defined over the complex numbers and its compact cohomology $H^i_c(X)$, under what conditions it is possible to compute its singular cohomology $H^i(X)$.
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65 views

Cohomology with coefficients in a commutative ring, how are the chain groups defined?

I have been studying a course in algebraic topology that follows Hatcher's textbook on the subject. I have some queries as to how certain things are defined. The first part of the text defines the ...
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0answers
28 views

Does additivity of (equivariant) cohomology hold at the algebra level?

The additivity property of many (co)homology theories is that if $X = \bigsqcup_{i \in I} X_i$ then $H^*(X) = \bigoplus_{i\in I} H^*(X_i)$. This is usually either an axiom of the theory, can be proven ...
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1answer
98 views

Isomorphism in cohomology is an isomorphism in homology

Let $f:X \to Y$ be a continuous map between topological spaces and $R$ some coefficients. From the universal coefficient theorem for homology we immediatly get, that if $H_*(f,\mathbb{Z})$ is an ...
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0answers
31 views

Uniqueness of the cohomological functor

This question is from the chapter 'Cohomology of Groups' by Atiyah and Wall in Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group. Theorem 1 on page 95 says that there is a ...
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1answer
28 views

A G-isomorphism of certain Hom groups

This question is from 'Cohomology of Groups' by Atiyah and Wall, p.95 of Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group and $A={\rm Hom}_{\mathbb Z}(\mathbb Z[G],X)$ where ...
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0answers
34 views

Transgression homomorphism on cohomology

I have one confusion about the transgression homomorphism which I found in two different books. I am unable to show that they really are same. The first description of transgression homomorphism I ...
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1answer
59 views

Vanishing of higher direct images of a composition

In a paper I am studying we have the following situation. Let $S$ be the spectrum of a Dedekind domain, and let $X$, $Y$ and $Z$ be scheme of finite type over $S$, where $X$ and $Y$ are smooths and ...
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2answers
75 views

Spaces with different homotopy type

I want to show that the spaces $ S^1 \vee S^1 \vee S^2$ and $S^1 \times S^1 $ do not have the same homotopy type. I calculated their homologies and cohomologies and they turn out to be equal. So I ...
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0answers
30 views

Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...