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

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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
83 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 ...
1
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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 ...
3
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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
74 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 ...
2
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1answer
53 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
24 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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0answers
31 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
43 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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30 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
59 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
37 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 ...
6
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1answer
104 views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
5
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0answers
48 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
126 views

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

I am trying to calculate the cohomology of $X = \mathbb R^2 \setminus \lbrace \mathbb Z \times \mathbb Z \rbrace = \lbrace (x,y) \in \mathbb R^2 : x \text{ and } y \not \in \mathbb Z \rbrace.$ ...
2
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2answers
64 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 ...
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0answers
79 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
votes
1answer
78 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 ...
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0answers
53 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
43 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
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1answer
58 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 ...
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0answers
22 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
63 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
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0answers
30 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
42 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 ...
1
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0answers
23 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
62 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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0answers
25 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
28 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
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1answer
69 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
18 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
53 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
66 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 ...
20
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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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0answers
35 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 ...
1
vote
2answers
54 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
59 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
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1answer
23 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 ...
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0answers
13 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 ...
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0answers
44 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
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1answer
93 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
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1answer
58 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 ...
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0answers
59 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 ...
4
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1answer
54 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 ...
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3answers
219 views

What's the point of spectra?

I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and ...
6
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1answer
124 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 ...
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44 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 ...
1
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1answer
66 views

Equivariant cohomology: $H^{*}_{S^{1}}(S^{2})$

We want calculate $H^{*}_{S^1}(S^2)$. We can choose two open sets $U= S^{2} \setminus p_{+}$ and $V = S^{2} \setminus p_{-}$, where $p_{+}$ and $p_{-}$ are the north pole and south pole of $S^{2}$. ...