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

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4
votes
1answer
103 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
3
votes
1answer
57 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
65 views

Intuition for chains and cochains

I'd like to get some "geometric," "physical," (or other form of) intuition for chains, cochains, and their relationship to integration on manifolds at an elementary level. In particular, it would be ...
2
votes
0answers
72 views

Relation between algebraic hyper de Rham cohomology and hodge theory in positive characteristic

I have recently been looking at algebraic de Rham cohomology of curves in positive characteristic. In particular, I am looking at when the sequence $$0 \rightarrow H^0(X,\Omega_X) \rightarrow ...
2
votes
0answers
35 views

uniqueness of Hopf invariant

(Hopf invariant, page 427 of A. Hatcher's Algebrac Topology): Let $f: S^m\longrightarrow S^n$ with $m\geq n$. We can form a CW-complex $C_f$ by attaching a cell $e^{m+1}$ to $S^n$ via $f$. The ...
0
votes
1answer
84 views

understanding into algebraic terms difference between homology and cohomology

my previous question understand quotient group was related to understanding of quotient group,i dont need to know too much detailed in group theore,just some part of algebraic topology,especially ...
1
vote
0answers
39 views

On the group cohomology $H^1(SL_2(\mathbb{Z}), \mathcal{M}(\mathbb{C}))$

Inspired by an answer to What is the intuition between 1-cocycles (group cohomology)?, one may wonder what are the meromorphic functions $f \in \mathcal{M}(\mathbb{C})$ for which there exists a ...
1
vote
0answers
48 views

Chains and cochains: integer versus real coefficients

Let a real, smooth manifold $M$ be given. For each non-negative integer $k$, let a singular $k$-cube on $M$ be a continuous mapping $c:[0,1]\to M$. Let $C_k(M,\mathbb Z)$ denote the set of formal ...
2
votes
0answers
36 views

How to compute the chi-y genus for a non-Kahler manifold?

I am a physicist and first-time poster so I will do my best to make this question clear. Apologies in advance if it's trivial. I am trying to compute the chi-y genus of the "Goldstein-Prokushkin ...
3
votes
2answers
154 views

de Rham cohomology of $\mathbb R^2 \setminus \mathbb Z^2 $

I am trying to calculate the cohomology of $X = \mathbb R^2 \setminus \lbrace \mathbb Z \times \mathbb Z \rbrace = \lbrace (x,y) \in \mathbb R^2 : x \text{ and } y \not \in \mathbb Z \rbrace.$ ...
1
vote
2answers
81 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 ...
0
votes
0answers
29 views

Morse cohomol. $\cong$ De Rham cohomol.

Are you aware of any short proofs for that fact (references)? And also for the fact that Morse cohomol. isomorphic to Singular cohomol. Also, a silly question: In the following thesis, he proves ...
1
vote
0answers
29 views

Why is it sufficient to prove for $H^0$

I am reading the corestriction homomorphism on cohomology. To prove a proposition (for example the composition of corestriction and restrcition is the multiplication by the index), it says that it is ...
1
vote
1answer
80 views

Apparent contradiction when using Mayer Vietoris

I am using the Mayer Vietoris sequence to compute the de Rham cohomology of a twice punctured plane. I computed it first by choosing my open cover to be $U=R^2-{pt}$ and $V=R^2-{pt}$ (the points are ...
0
votes
0answers
21 views

Show that $con_H^g$ is the identity map

Let $G$ be a group,$H$ a subgroup of $G$ and $A$ be $G$-module. Let $g \in G$ and $H^g=g^{-1}Hg$. Let $Z^2(G,A)$ be the set of $2$-cocycles. Given $f \in Z^2(H,A)$, let $f^g \in Z^2(H^g,A)$ be defined ...
1
vote
0answers
90 views

Derived push-forward of projective sheaf

Let S,X be schemes and $s \in S$ be a closed point. Let $D(X)$ be the derived category of complexes of sheaves. Let $$i_s: X \cong {s} \times X \hookrightarrow S \times X$$ be the natural embedding. ...
0
votes
2answers
157 views

Exercises - “From calculus to cohomology”

I am reading Madsen's book From calculus to cohomology and I've found it doesn't have any (explicit) exercises at the end of each section. I'd like to know a few books where I can find some problems ...
2
votes
0answers
121 views

Cup Product Structure on the n-Torus

We know that The $k^{th}$ homology of the n-torus $(S^1)^n$ is generated by $\bigotimes_{i\in I}\lambda_i$ where $\lambda_i$ generates the first homology of the $i^{th}$ copy of $S^1$ and ...
1
vote
0answers
45 views

Compactness and Poincare duality

I am reading Appendix B in Fulton's Young Tableaux about Borel-Moore homology. In particular, I'd like to understand why for compact manifolds the Borel-Moore homology groups are isomorphic to ...
2
votes
2answers
64 views

Inductive definition of group cohomology?

At the start of Atiyah and Wall's section on group cohomology (in the Cassels-Frhlich collection of Algebraic Number Theory notes) they, of course, define group cohomology (actually, a 'cohomological ...
3
votes
1answer
77 views

Dolbeault cohomology and analytic regularity

Let $M$ be a complex analytic $n$-manifold. The Dolbeault cohomology complex is defined using a quotient space of smooth differential forms. My question is : would it make a big difference if we were ...
0
votes
0answers
26 views

The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
0
votes
0answers
47 views

When is $\mathcal{H}om$ functor exact in the category of presheaves

Let $C$ be a projective $\mathbb{C}$-scheme of pure dimension $1$. Suppose that $C$ is local complete intersection in $\mathbb{P}^3$. Let $C_1$ be an irreducible component of $C$, also of pure ...
0
votes
1answer
25 views

Why $\operatorname{im} \delta_{-1}$ contains two functions and $f\in C^{0}(X, \mathbb{F}_2)$ is the characteristic function?

In the paper, on page 21, line 15-20. It is said that $B^0=\operatorname{im} \delta_{-1}$ is the one dimensional space containing two functions and $f\in C^{0}(X, \mathbb{F}_2)$ is the characteristic ...
0
votes
1answer
135 views

cohomology groups of K(Zp x Zp, 1)

I have a question regarding the cohomology groups of the Eilenberg-MacLane space $K(\mathbb{Z}_p \times \mathbb{Z}_p,1)$. For $n$ > $2$, is there a way to show that $H^n(K(\mathbb{Z}_p \times ...
2
votes
1answer
65 views

Exact and closed forms with a vanishing Riemann tensor

I need a result to prove that an closed form is equally exact. I work under the assumption that the Riemann tensor vanishes everywhere on the manifold. (It is the context of general relativity.) The ...
0
votes
0answers
78 views

Exact sequences and spectral sequences

We have the well-known theorem for cohomological spectral sequences as follows: Theorem: Let $(E_r , d_r )$ be a third quadrant spectral sequence and let $E^{p,q}_2‎\Rightarrow‎ H^n(Tot(M)$. a) If ...
28
votes
0answers
510 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
5
votes
1answer
67 views

Show that image of $res$ lies in $H^n(H,A)^{G/H}$

Let $G$ and $G^{\prime}$ be groups, $A$ and $A^{\prime}$ be $G$-module and $G^{\prime}$-module respectively, $C^n(G,A)$ be set of all maps from $G \times \cdots \times G$ ($n$ times) to $A$, $d_n ...
6
votes
1answer
217 views

Cup Product Structure on the Projective Space

I am reading about cup products and am stuck on this exercise in Hatcher (3.2.5). Taking as given that $H^*(\mathbb{R}P^\infty,\mathbb{Z}_2)\simeq\mathbb{Z}_2[\alpha]$, how does one show ...
2
votes
0answers
73 views

Hochschild Serre Spectral sequence

I am reading Hochschild Serre Spectral sequence from Group Theory I by M Suzuki Page 213. I will be thankful to you if you make me understand following points to me. Following is the sequence $0 ...
5
votes
0answers
72 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
4
votes
0answers
164 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
923 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
78 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
39 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
159 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
356 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
74 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
123 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
67 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
146 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
74 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
112 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
88 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 ...