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

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1answer
62 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 ...
4
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1answer
92 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
21 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 ...
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1answer
43 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
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1answer
39 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
96 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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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
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1answer
38 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
37 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 ...
5
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1answer
118 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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460 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 ...
2
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0answers
35 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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78 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 ...
2
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0answers
36 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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38 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 ...
4
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1answer
92 views

When does cohomology take pullbacks to pushouts?

I've encountered a simple situation where one has a pullback diagram of topological spaces and taking cohomology takes it into what I believe is a pushout diagram in the category of rings. I'm not ...
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1answer
32 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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52 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 ...
3
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1answer
77 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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40 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 ...
2
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1answer
81 views

Sheaf cohomology of $\mathbb{P}^3$

Let $\mathbb{P}$ denote the projective space over $\mathbb{C}$. In some lecture notes I found the claim that $$ h^0(\mathbb{P}^3, \mathcal{O}(2)) = 10 $$ Do you know why this is the case? In ...
2
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0answers
34 views

map to product of eilenberg-maclane spaces

Given a space $X$, and an Eilenberg-MacLane space $K(G,n)$ (hereafter referred to as $K$), and two maps $f: X \to K$ and $g:X \to K$, let $f \times g:X \to K \times K$ map $x \in X$ to $(f(x),g(x))$. ...
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49 views

Intersection form and poincaré duality

Let $ M $ be a $2n$-dimensional compact connected oriented smooth manifold and let $A$, $B$ be two $n$-dimensional submanifolds that intersect transversally. Denote by $A \cdot B$ the sum of the ...
3
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1answer
47 views

Global sections of a twisting of a structure sheaf of a projective scheme

Let $X$ be a projective Noetherian scheme over $\mathbb{C}$. Is it true that $H^0(\mathcal{O}_X(-t))=0$ for any $t>0$?
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37 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
2
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1answer
64 views

Vanishing of $R^1f_*\mathcal O_X$

I am probably missing something obvious here, but none the less, here goes: Is the following statement (or perhaps some minor modification of it) true and if so, why: $R^1f_*\mathcal O_X = 0$ for a ...
4
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0answers
43 views

what can you say about the degree of $f:\mathbb{C}P^n \to \mathbb{C}P^n$

Any thoughts on this problem: If $M$ and $N$ are simply-connected, $n$-dimensional manifolds, then $H^n(M;\mathbb{Z}) \cong \mathbb{Z} \cong H^n(N;\mathbb{Z})$. A map $f:M \to N$ induces a map ...
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57 views

Making modular representation theory and cohomology 'compelling' and 'accesible'

I'm currently putting together an application for a dissertation completion fellowship offered through my university. A part of the application includes 500-1000 words describing my dissertation. ...
3
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2answers
142 views

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

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

Cochains: terminology

Let a real, smooth manifold $M$ be given. Let $C_k(\mathbb Z, M$) denote the set of $k$-chains with integer coefficients, and let $C_k(\mathbb R, M)$ denote the set of $k$-chains with real ...
5
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0answers
98 views

Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
4
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1answer
95 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 ...
2
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0answers
69 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 ...
3
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1answer
64 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
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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 ...
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1answer
78 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 ...
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0answers
36 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 ...
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0answers
47 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 ...
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0answers
33 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 ...
2
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0answers
93 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 ...
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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 ...
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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 ...
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1answer
78 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 ...
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0answers
20 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 ...
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0answers
78 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. ...
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2answers
121 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 ...
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1answer
1k views

Cohomology of projective plane

How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
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42 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 ...
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2answers
58 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
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1answer
75 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 ...