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

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1answer
34 views

Does the cup product on de Rham cohomology induce a nondegenerate bilinear form?

I have small issue I came across in the following. Suppose $M$ is a compact, oriented manifold of dimension $4n+2$. I want to prove that the de Rham cohomology group $H^{2n+1}(M)$ are even ...
6
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1answer
160 views

Prove Poincare duality theorem with Morse theory.

First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
0
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2answers
62 views

Abelian category without enough injectives

What is an example of an abelian category that does not have enough injectives? An example must exist, but I haven't been able to find one. If possible, a brief explanation of why the abelian ...
1
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2answers
87 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 ...
2
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2answers
66 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
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0answers
51 views

cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
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0answers
24 views

What maps of $k$-algebras $A\to B$ induce finite maps $\mathrm{Ext}_B^*(k,k)\to\mathrm{Ext}_A^*(k,k)$?

Let $k$ be an algebraically closed field, and let $A$ and $B$ be finitely generated $k$-algebras. A map $\varphi:A\to B$ of $k$-algebras induces a map ...
2
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2answers
49 views

Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...
2
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0answers
9 views

Connecting homomorphism in the Gysin sequence

Let $j : SO(2n) \to SO(2n+1)$ be the standard subgroup embedding and let $Bj : BSO(2n) \to BSO(2n+1)$ be the induced fibration obtained by factoring the universal $SO(2n+1)$-bundle by the subgroup ...
0
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1answer
40 views

Hard Lefschetz Thereom and the Cohomology of Flag Varieties

Let $G$ be a compact connected connected Lie group $G$, and $B$ a Borel subgroup containing a maximal torus. Moreover, let $F = G/B$ be the associated flag manifold. Now $F$ is a compact Kahler ...
0
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1answer
27 views

smash product of Eilenberg-Maclane spaces

Let $G$ be an abelian group and $K_n=K(G,n)$ be the Eilenberg-Maclane space. How to obtain $K_m\wedge K_n$ is $(m+n-1)$-connected? (Hatcher's book page 404)
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0answers
55 views

Showing Grothendieck's Vanishing Theorem provides a strict bound

The following result is due to Grothendieck: If $X$ is a noetherian topological space of dimension $n$, then for all $i>n$ and all sheaves of abelian groups $\mathscr{F}$ on $X$, we have ...
1
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0answers
28 views

Meyer-vietoris sequence to compute the compact cohomology for Möbius strip

How do you use Meyer-vietoris sequence to compute the compact cohomology for Möbius strip without the bounding edge? Please give detail math. In particular explain how inclusion map is used. On page ...
1
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0answers
29 views

Rearding notation of (Relatively)Projective/ (Relatively)Injective in Group cohomology

I am reading Group cohomology from Serre's Local Fields. I got confused with the notation he used... We know that : $A$ is Projective module if $Hom_R(A, \_)$ is exact $A$ is Injective module if ...
4
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3answers
349 views

How do I know when a form represents an integral cohomology class?

Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class. I would like to ...
1
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1answer
42 views

Showing the image of $H^j(X;\mathbb C^\times)$ lies in the torsion subgroup of $H^{j+1}(X;\mathbb Z)$

Let $X$ be a (compact, if necessary) topological space. Then from the short exact sequence of constant sheaves $$ 0 \to \mathbb Z \to \mathbb C \to \mathbb C^\times \to 0 $$ we have a connecting ...
1
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1answer
30 views

2-forms on $S^2$

I've read that the group $H^2_{dR}(S^2)=\mathbb{R}$. If I'm not wrong, this implies that one can build closed 2-forms that are not exact. Can somebody show me an example, please? Thanks!
2
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1answer
72 views

Some questions about cellular homology and cohomology

Consider the CW structure on $\mathbb{RP}^n$ given by one cell in every dimension. This gives rise to the cellular complex $C_\bullet(\mathbb{RP}^n)$ which is generated by a single element $c_i$ for ...
6
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0answers
228 views

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
1
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1answer
23 views

Supplement for reading Group cohomology from Serre Local Fields

I am doing a reading course on Group cohomology... I am supposed to start reading Group cohomology part in Serre's Local fields Book ...
2
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0answers
38 views

Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
0
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0answers
47 views

Non-existance of a neutral element for the cup product

I know that if $X$ is a space and $R$ a commutative unitary ring, the cup product $$\smile : H^k(X; R) \times H^l(X;R) \rightarrow H^{k+l}(X;R)$$ has as a neutral element the cohomology class in ...
1
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1answer
32 views

Quantum Cohomology of Affine Toric Varieties

I would like to know whether quantum cohomology rings of affine toric varieties have been calculated, if this is possible. Does anyone have a relevant paper they could refer me to? I have seen it ...
0
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1answer
39 views

Equality in de Rham cohomology

Let $U_1,U_2,...,U_r$ be open sets in $\mathbb{R}^n$ such that $U_i\cap U_j =\emptyset$ for all $i \neq j$. Then prove, $H^k_{dR}(\bigcup_{i=1}^{r} U_i)=\bigoplus_{i=1}^{r} H^k_{dR} (U_i)$
7
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1answer
141 views

Does the rank of homology and cohomology groups always coincide?

Let $(C_i)_{i \in \mathbb{Z}}$ be a chain complex of free abelian groups. Does the rank of the homology and cohomology groups of $(C_i)_{i \in \mathbb{Z}}$ always coincide, i.e. is ...
1
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0answers
31 views

Cup products and cross products

So, I am trying to compute some products on chains and their duals, but I have difficulties in understanding some operations. The cup product of cochains is quite easy to understand, especially when ...
0
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1answer
49 views

Simple example of $X$ with torsion in $H^1(X,\mathbb{Z})$?

Question: Is there a simple example of a space $X$ possessing torsion in its first integral cohomology group $H^1(X,\mathbb{Z})$? For reasonable spaces $X$, e.g. CW-complexes, one has ...
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0answers
24 views

Looking for Proof of the Order of a quandle cohomology group for an n-fold dihedral

Does anyone know where the proof that states that the order of a quandle cohomology group for an n-fold dihedral quandle is n is located? I have been told it can be found in one of ...
3
votes
1answer
26 views

Generalisation of cochain complexes and “curvature”

Someone has mentioned to me that generalizations of co-chain complexes and their cohomology have been studied, where instead of $d^2 = 0$ we have something like $d^2 \alpha = q \alpha $, which is ...
2
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1answer
65 views

The cohomology ring of the nerve of a category associated to a vector space

Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are ...
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0answers
53 views

Prove the Weyl's complete reducibility Theorem on finite-dimensional $\mathfrak{g}-modules$ by Kostant's $\mathfrak{n}$-cohomology result

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)). But I have no idea about its proof. Any ...
2
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1answer
44 views

An isomorphism between homology and cohomology with $\mathbb{Z}_2$ coefficients

In the proof of a theorem we did in a class (namely: if $M$ is an odd-dimensional, closed manifold, then $\chi(M)=0$), there's the following step: $$H_k(M;\mathbb{Z}_2)\cong H^k(M;\mathbb{Z}_2)$$ ...
1
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1answer
40 views

First cohomology group of direct products

Let $p$ be a prime number and H be a finite group with $|H|=p-1$ and consider $\varphi: H \times Z_{p^k} \rightarrow Aut(Z_{p^k})$ as a non-trivial action of $H \times Z_{p^k}$ on $Z_{p^k}$ such that ...
1
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0answers
50 views

how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...
4
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1answer
51 views

Simplicial homology of the skeleton of a simplex

Let $n$ and $k$ two natural numbers. We consider the (abstract) simplicial complex $K$ on $n$ vertices $v_1,\dots,v_n$ and such that a subset of $\{v_1,\dots,v_n\}$ is a face of $K$ if and only if it ...
8
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1answer
163 views

Does de Rham theorem hold for manifolds with boundary?

I am following the J.Lee's book "Introduction to Smooth Manifolds", 2nd ed., page 480-486 to learn the de Rham theorem. It is proven on manifolds without boundary, which makes me curious about whether ...
3
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0answers
54 views

Geometric Interepretation of $\mathbb{G}_a$-torsors

Let's fixed a locally ringed space $(X,\mathcal{O}_X)$ (although, this should apply to any ringed topos, but I haven't thought that through). In fact, if it's helpful, you can assume that $X$ is a ...
3
votes
1answer
58 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
1
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2answers
114 views

Cohomology Ring of Klein Bottle over $\mathbb{Z}_2$

I am trying to show that the cohomology ring of the Klein bottle with $\mathbb{Z}_2$ coefficients is $H^*(K,\mathbb{Z}_2) \cong \mathbb{Z}_2[x,y]/(x^3,y^2, x^2y)$. What I know: ...
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0answers
41 views

Tensor product with an L on top

Looking at the definition of the Kunneth morphism in SGA4, XVII, 5.4.1.4, there is the notation $$Rf_*K \overset{\mathbb{L}}{\boxtimes}_{\mathcal{A}_0} Rg_* L \rightarrow Rh_*(K ...
4
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0answers
36 views

Compute the induced map on $\mathbb{CP}^n$

Let $d>0$ and $f:\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$ be given by $f(z_0,...,z_n)=(z_0^d,...,z_n^d)$. Let $F:\mathbb{CP}^n \rightarrow \mathbb{CP}^n$ be the induced map by $f$. Compute ...
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0answers
45 views

Relative cohomology of a vector space module non-zero vectors

I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer ...
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5answers
1k 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 ...
5
votes
2answers
62 views

history and/or motivation for cohomology in class field theory

I am currently learning (local) class field theory via group cohomology with Milne's notes. I have a number of questions about using group cohomology to prove the main statements of class field ...
3
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1answer
53 views

Trivial Cohomology Group->Lower-Dimensional Homotopy?

Calculating the (de-Rham) cohomology of a tee connector (Picture), I got $H^0=R,H^1=R^2,H^2=0$. Furthermore, just from looking at it, I assume the tee connector is homotopic to a circle with an arc ...
3
votes
1answer
55 views

On which groups can homomorphisms be defined subgroup-locally?

Let $G$ be a group and suppose that $G$ is the set-theoretic union of a collection $\mathcal U$ of subgroups. I will call such a collection a "covering" of $G$. Suppose that for each $L \in \mathcal ...
9
votes
1answer
193 views

Relative de Rham Cohomology is Homotopy Invariant

Suppose $ f:N\rightarrow M$ is a smooth map between two manifolds. Relative de Rham cohomology is defined through the complex $ \Omega^{q}(f)=\Omega^{q}(M)\oplus\Omega^{q-1}(N)$ with ...
1
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1answer
34 views

Is $H^0(S^0;G)\simeq G\oplus G$ or $G$?

In the article on topospaces for the (co)homology of spheres, it says $H^0(S^n,G)\simeq H^n(S^n,G)\simeq G$. Is this true when $n=0$? I think not, for if we view $S^0$ as the union of two ...
0
votes
1answer
27 views

Degree of an induced map on $\mathbb{CP}^n$

Let $r :\mathbb{C}^{n+1} \rightarrow \mathbb{C}^{n+1} $ be the map $r(z_0, z_1,\ldots, z_n)=(-z_0, z_1,\ldots, z_n)$. $r$ induces a map $\bar r : \mathbb{CP}^n \rightarrow \mathbb{CP}^n $. What is the ...
0
votes
1answer
45 views

Why is $H^2(\mathbb{R}P^2,\mathbb{Z})\simeq\mathbb{Z}_2$?

Why is the second cohomology group of $X=\mathbb{R}P^2$ with $\mathbb{Z}$-coefficients $\mathbb{Z}_2$? We can put the usual $\Delta$-structure on $X$ with two vertices, three $1$-simplices, say $a$, ...