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

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4
votes
0answers
143 views

Top de Rham cohomology

I just realized that I never really understood why $H_{dR}^n(M, \mathbb{R}) = \mathbb{R}$ if $M$ is compact and $H_{dR}^n(M, \mathbb{R}) = \{0\}$ if $M$ is not compact (provided that's true?). I'm ...
7
votes
2answers
843 views

how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
3
votes
1answer
77 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
5
votes
0answers
37 views

Computing cohomology of product space with product-ring coefficients

I'm interested in the following problem: Let $X$ and $Y$ be finite CW complexes and $R$ and $S$ rings. Suppose you are given the cohomology rings $H^* (X; R)$ and $H^* (Y; S)$. Is there an easy way ...
0
votes
1answer
59 views

how can I show $H^1(g , Hom_C(g,M))=0$?

For a simple Lie algebra $g$ and a finite dimensional vector space $M$ with a trivial $g-$action, how can I show $H^1(g , Hom_C(g,M))=0$?
7
votes
0answers
151 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
2
votes
1answer
339 views

Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
2
votes
1answer
70 views

About existence of Morse functions

Let's consider 4-manifold $M$, $\partial M = \partial M_1 + \partial M_2 = S^1 \times S^2 + \mathbb{RP}^3$. Is it true that there exist a Morse function $$f\colon M^4 \to [0,1],\quad f^{-1}(0) = ...
3
votes
1answer
121 views

Cohomology groups

I have some questions. 1) I tried to compute the cohomology group of $S^3$ with coefficients in $\mathbb{Z}/2\mathbb{Z}$ but I don't know if my result $$ H^k(S^3,\mathbb{Z}/2\mathbb{Z}) = ...
2
votes
0answers
65 views

Derived functors and coboundary operator

I understand that one can define the cohomology of an object $A$ in terms of a complex (non-zero in positive degrees) in some Abelian category, together with differentials, such that the composition ...
0
votes
0answers
27 views

Problem with cohomology (II)

Let $G$ be a group and $K$ be subgroup of $G$, Let $A$ be $G$ module. Let $X=\{X_n\}$ be free resolution of $\mathbb{Z}$. Let $\delta$ represents collectively the homomorphism induced in Hom sequence ...
0
votes
1answer
46 views

Problem with cohomology (I)

I have some doubts regarding cohomology. As title suggests I will ask these one by one. Let $G$ be a group and $A$ be $G$-module. Let $C^n(G,A)$ denote the set of all maps from $G \times \cdots ...
1
vote
0answers
26 views

Conservativeness on a graph

I'm trying to build a conservative vector field out of something smaller than $\mathbb{R}^2$ to understand how the "conservative" property of differences-of-scalar-fields leads to Green's theorem. (In ...
1
vote
2answers
127 views

Cohomology groups for the following pair $(X,A)$

Let $X=S^1\times D^2$, and let $A=\{(z^k,z)\mid z\in S^1\}\subset X$. Calculate the groups and homomorphisms in the cohomology of the exact sequence of the pair $(X,A)$. I know that theorically one ...
5
votes
0answers
69 views

Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where ...
1
vote
0answers
27 views

Simple question on splitting of cohomology groups.

From the exponential exact sequence, I have $$ 0 \rightarrow H^2(X,\mathbb{C})/H^2(X,\mathbb{Z})\rightarrow H^2(X,\mathbb{C}^\times) \rightarrow Tor(H^3(X,\mathbb{Z})) \rightarrow 0. $$ for some ...
3
votes
1answer
98 views

Global section of pull-back of structure sheaf of projective scheme

Let $X$ be a smooth projective variety and $Z_1, Z_2$ two smooth projective divisors in $X$. Is it true that the natural restriction morphism from $H^0(\mathcal{O}_X(-Z_1-Z_2))$ to $H^0(\mathcal{O}_X ...
2
votes
1answer
85 views

Flag varieties and representation theory

I've recently been reading about flag varieties and their cohomology. I'm mainly interested in representation theory, and I've heard that flag varieties are important objects, especially in Lie ...
1
vote
1answer
58 views

Differential forms as functionals on curves

Please give me a reference to a book or lecture notes where the following stuff is studied. Let $M$ be a Riemann surface with boundary $\partial M$ (but not necessarily, any smooth $n$-dimensional ...
2
votes
0answers
51 views

Definition of the relative de-Rham cohomology and its generalization

Let $M$ be a smooth manifold and $N$ be its smooth submanifold. We say that two closed forms $\omega_1$ and $\omega_2 \in \Lambda^k(M)$ are equivalent if their difference is an exact form from ...
1
vote
1answer
84 views

Cohomological ($p$-)dimension of a pro-$p$ group

I have a question concerning the cohomological dimension and $p$-dimension of a pro-$p$-group. Let's first recall the definitions of that The cohomological dimension $cd \ G$ of a pro-finite group ...
1
vote
1answer
76 views

simple question about cohomology group

Let's consider compact 4-manifold $M^{4}$ and point $P \in M$. Then (use Mayer-Vietoris) inclusion $i\colon M\setminus P \to M$ induce isomorphism $i^{*}\colon H^2(M) \to H^2(M\setminus P)$. Let's ...
1
vote
0answers
47 views

Relative de Rahm cohomology computation for two disjoint circles embedded in R^2

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
7
votes
1answer
91 views

Finite generation of Tate cohomology groups

Let $G$ be a finite group, and let $F$ be a complete resolution for $G$. In other words, $F$ is an acyclic chain complex of projective $\mathbb{Z}G$-modules together with a map ...
1
vote
1answer
59 views

An exact homology sequence associated with a principal SO(n) bundle

Suppose $P$ is a principal $SO(n)$ bundle, X is its base space. Why is there an exact sequence in homology groups $$ 0 \to H^1(X;\mathbb{Z}_2) \to H^1(P;\mathbb{Z}_2) \to H^1(SO(n);\mathbb{Z}_2)\to ...
3
votes
1answer
97 views

Cech cohomology and cohomology of a category : a cluster of questions.

I apologize in advance : what follows is a bit of a mess. Also, I think it might be a big tautology, but i don't see it yet. My question is about the rapport of Cech cohomology and cohomology of a ...
1
vote
0answers
30 views

Weight space for a finite-dimensional $\mathfrak{g}-$module $M$.

Let $\mathfrak{g}$ a semisimple Lie algebra, $M$ finite-dimensional $\mathfrak{g}-$module, $\mu\in\mathfrak{h}^*_{\mathbb{Z}}$ and $s_i$ simple reflection such that ...
2
votes
0answers
37 views

Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
8
votes
0answers
100 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
2
votes
0answers
31 views

What is Kadison's process about cocycles?

My teacher told me the Kadison's process(may be not this ward, it is just my translation ) can make a 2-cocycle turn to be a cocycle(i.e.,derivation). But I can not find it in the internet. Thanks a ...
2
votes
1answer
345 views

First proof of Poincaré Lemma

I know that a way of proving Poincare lemma is to use the homotopy invariance and contractibility of the Euclidean space. Is there is a way of doing it directly (without using the contractibility of ...
1
vote
1answer
293 views

The Poincare Lemma for Compactly Supported Cohomology

I´m reading the proof of The Poincare Lemma for Compactly Supported Cohomology there is a part in the proof that said in the text book Bott and Tu: $d \pi_{\ast} = \pi_{\ast} d$ in other words, ...
1
vote
0answers
66 views

Easy characterization of Cohomology in an Abelian Category

It should be quite an easy question and probably there's also a certain degree of intrinsic silliness in it, but still... Let $\mathcal{C}$ be an abelian category and let $C(\mathcal{C})$ be the ...
4
votes
0answers
121 views

Functors that are the homology of a chain complex

Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, ...
0
votes
1answer
107 views

Surfaces of genus g

The problem: give maps $f:\Sigma_{g}\longrightarrow\Sigma_{h}$ not homotopic to a constant map with $0<g<h$. Any idea would be helpful.
4
votes
0answers
69 views

Applications of Microfunctions

Can anyone suggest good (a) uses/applications or (b) construction of micro-functions (introduced by Mikio Sato in 1971) in analysis? I am trying to understand the subject better. Suggestions of ...
8
votes
2answers
104 views

Cohomological definition of the Chow ring

Let $X$ be a smooth projective variety over a field $k$. One can define the Chow ring $A^\bullet(X)$ to be the free group generated by irreducible subvarieties, modulo rational equivalence. ...
3
votes
1answer
83 views

Extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$

Given that $H^2(\mathbb{Z}_n,\mathbb{Z})=\mathbb{Z}_n$, it follows that up to equivalence there should be $n$ extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$, one for each cohomology class. I'd like to ...
0
votes
0answers
45 views

First cohomology of a Galois group with finite base field

Let $l/k$ be a (may be infinite) galois extension with galois group $G$ and $k$ a finite field with size $q$. Also $k$ and $l$ are given the discrete topology. $G$ is given the Krull topology. Then ...
7
votes
1answer
133 views

Computing the action of $S_3$ on $H^n(\mathbb{Z}_3,\mathbb{Z})$

Let $G=S_3$ and let $H$ be the Sylow $3$-subgroup in $G$. If $\mathbb{Z}$ is the trivial module, then it can be shown that $$H^n(H,\mathbb{Z})=\begin{cases}\mathbb{Z}&n=0\\0&n\text{ ...
0
votes
1answer
73 views

Non abelian $H^1(G,A)$ problem.

Let $G,A$ be groups. We do not assume that $A$ is abelian. For $f,g\in Z^1(G,A)$, we write $f\backsim g$ if there is an $a\in A$ such that $g(x)=a^{-1}f(x)\ ^xa$ (we use the pre-exponential notation ...
7
votes
1answer
116 views

Strange case of Serre's duality

$\newcommand{\O}{\mathcal{O}}$ Let $X$ be a smooth projective curve and $D$ and effective divisor on it. The normal bundle of $D$ is defined as $$ \O_D(D)\; = \; \O_x(D)\;\otimes_{\O_X}\, \O_D$$ where ...
2
votes
1answer
55 views

confused, Universal Coefficient Theorem (cohomology)

This is bad, but I was applying the UCT to a small complex and didn't seem to work. Namely the chain complex $0 \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow 0$ where the nonzero map is, ...
6
votes
1answer
469 views

Long exact sequence for cohomology with compact supports

Related to my previous question here. Let $X$ be a topological space and let $H_c^{\bullet}(X)$ denote its singular cohomology with compact supports (rational coefficients). Let $U$ be an open subset ...
3
votes
0answers
80 views

Tangent space of a moduli space.

Let $X$ be a compact Riemann surface with genus $2$ and $M^2$ the moduli space of stable principal $SL(2)$-bundles of rank $r$. We know that $M^2$ is a complex projective variety of dimention ...
3
votes
1answer
423 views

Understanding cohomology with compact support

I am trying to understand the definition of (singular) cohomology with compact supports. My understanding of singular cohomology goes like this. Let $X$ be a topological space. Define the singular ...
1
vote
1answer
138 views

cohomology of Eilenberg-Maclane space

In line 5, Page 394 of Allen Hatcher's book Algebraic Topology, it is claimed that $H^n(K(G,n);G)=Hom(H_n(K(G,n),\mathbb{Z});G)$ for any abelian group $G$. How to get it? I have tried but cannot ...
2
votes
1answer
45 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...
0
votes
1answer
31 views

References on “relative Lie groups”

I'm trying to read Deligne's Formes modulaires et représentations $\ell$-adiques. In section 2, he briefly goes over a number of facts about (complex-analytic) elliptic curves in a relative setting. I ...
5
votes
2answers
74 views

Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.

Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ if $m < 0$. This statement came up in an algebraic geometry text with no explanation ...