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

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0answers
29 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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1answer
32 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 ...
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1answer
20 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 ...
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1answer
43 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 $a$, ...
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2answers
44 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 ...
2
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1answer
50 views

Wedge Sum of Two Spheres Homotopy Equivalent to a Compact Manifold?

Let $X=S^2$v $S^2$ (wedge sum). The homology groups are $H_0(X,\mathbb{Z})= \mathbb{Z}$, $H_1(X,\mathbb{Z})= 0$, and $H_2(X,\mathbb{Z})= \mathbb{Z} \oplus\mathbb{Z}$. I can see that $X$ is not ...
2
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1answer
41 views

Cohomology of classifying spaces of exceptional Lie groups

Given an exceptional Lie group $G$ and maximal torus $T$ thereof, the inclusion $T \hookrightarrow G$ induces a map $BT \to BG$ of classifying spaces and a cohomological pullback $$H^*(BG) \cong ...
3
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1answer
54 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 ...
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0answers
34 views

Is this cohomology isomorphic to De Rham Cohomology?

Let $(M,g)$ be a Riemannian manifold. Put $d^{*}= *d*$ where $*$ is the Hodge $*$ operator. So $d^{*}\circ d^{*}=0$. Then it introduce a (c0)homology. What is a relation between this ...
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0answers
29 views

Book Suggestion - Complex algebraic surfaces

I am studying for an exam of algebraic geometry, in particular, I am dealing with ruled surfaces and numerical invariants, rational surfaces, Castelnuovo's Theorem and its application. I am reading ...
3
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0answers
57 views

De Rham cohomology of the pointed plane

i try to work out some examples for de Rham cohomology, but i have some problems: I want to figure out what $H^k(\mathbb{R}^2\setminus\{0\})$ is and want to generalize this to arbitrary finite points ...
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0answers
84 views

Show $\mathbb{CP^2/CP^1}$ is not a retract of $\mathbb{CP^4/CP^1}$.

So I have shown that the natural projection $\pi: \mathbb{CP^n}\rightarrow \mathbb{CP^n/CP^k}$ induces a monomorphism $\pi^*:H^*(\mathbb{CP^n/CP^k},\mathbb Z)\rightarrow H^*(\mathbb{CP^n},\mathbb Z) ...
3
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0answers
40 views

When is a graded ring the cohomology ring of a CW-complex?

Let $A^*$ be a graded-commutative ring with $A^n = 0$ for sufficiently large $n$ and each $A^n$ finitely generated. When does there exist a finite CW-complex $X$ with $H^*(X) \cong A$ as graded rings? ...
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1answer
27 views

$H^1(X) = [X,\mathbb{T}]$?

This is a stupid question, but here goes. I have a compact Hausdorff space $X$, and I am talking about $[X,\mathbb{T}]$, the group of homotopy classes of maps $X \to \mathbb{T}$, where $\mathbb{T}$ ...
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1answer
17 views

not complete solution in computation of singular cohomology

I know how to compute singular cohomology of single point. Then I want to compute cohomology of $\mathbb{Z}$. Consider $C_{k}(\mathbb{Z})=\oplus_{i\in Z} C_{k}(point)$. $$\text{Hom}(A\oplus B, G)= ...
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0answers
52 views

De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
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0answers
18 views

Comparison between Eilenberg-Steenrod excision and Brown representability excisive

One of the Eilenberg-Steenrod axioms for unreduced cohomology is excision, which states that $H^n(X,A)\cong H^n(X\setminus U,A\setminus U)$, for good subspaces such as when $\overset{\circ}U\subseteq ...
5
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1answer
79 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
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29 views

Dimensions of the cohomology groups of certain complicated space

Let contruct the space $X$. We take the complex projective space $\mathbb{C}P^2$, pick two points $p_1, p_2 \in \mathbb{C}P^2$ and remove two small, disjoint, open $4$-balls $B_j$ centered at $p_j$. ...
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0answers
55 views

de Rham cohomology group of $\mathbb{CP}^1$

Prove the de Rham cohomology group of the projective space $H^0(\mathbb{CP}^1, \mathbb{C})$ is isomorphic to $\mathbb{C}$ and $H^1(\mathbb{CP}^1, \mathbb{C})=0$.
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1answer
26 views

Obtaining Chain Complex from a Cochain Complex

In this question: Constructing a cochain complex out of a chain complex , palio asked how to construct a co-chain complex when given a chain complex as well as how to go in the opposite direction, ...
0
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1answer
41 views

Cohomology Group

I am new to the area of cohomology, and I have got stuck on the definition of the abelian group $C_n(G, N)$ of $n$-cochains of $G$ with coefficients in $N$. I quote here from Anthony W. Knapp's ...
3
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36 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary ...
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1answer
64 views

Decomposition of cohomology group on $S^{n}$

If we have decomposition of cohomology group on $S^{n}$ it looks like $H^{n}(S^{n})=H^{n}(S^{n})_{+}\oplus H^{n}(S^{n})_{-}$, where $H^{n}(S^{n})_{\pm}$ cohomology of invariant or anti-invariant $n$ ...
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1answer
185 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 ...
3
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1answer
39 views

Well-definedness of a coboundary map between a reduced $L^2$ de Rham cohomology group and a relative cohomology group

I'm working right now with this paper of Carron. And I think I'm stuck at a relatively simple question. On page 11 he is defining a coboundary map $b : H^k_{2, \text{reduced}}(M - K) \to H^{k+1}(K, ...
1
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1answer
50 views

Poincare lemma via Lie derivative

I found such a beautiful proof of Poincaré lemma here (in Russian): Let $B$ be a star-like neighborhood of 0 in $\mathbb{R}^n$ and $r=\sum x^i \frac{\partial}{\partial x^i}$. Then Lie derivative ...
2
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1answer
35 views

Wedge product of basis elements of cohomology

Let $M$ be a compact, connected, oriented 4-manifold without boundary. If $H^2(M)\cong \mathbb{R}^2$ and I have a basis $\{[\omega_1],[\omega_2]\}$ for $H^2(M)$, is it the case that $[\omega_1\wedge ...
2
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1answer
39 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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1answer
65 views

De Rham cohomology for $\mathbb{R^2}$

De Rham cohomology groups for $\mathbb{R^2}$. $H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R}$ since $Z^{0}(\mathbb{R}^{2})$ is the one dimensional space of locally constant functions on $\mathbb{R}^{2}$ and ...
1
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1answer
43 views

Prove that $H^{2}(S^{2})\neq 0$

Prove that $H^{2}(S^{2})\neq 0$ Suppose $\omega$ is an area form of $S^{2}$. An arbitrary two form on $S^{2}$ is closed as if $f(x,y)dx\wedge dy\in\Omega^{2}(S^{2})$ then $d(f(x,y)dx\wedge dy)=0$. I ...
2
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2answers
73 views

Total dimension of the cohomology of a homogeneous space (or of a graded Tor)

I want to calculate the cohomology ring with rational coefficients of a homogeneous space, but would be happy enough to know its total dimension. Let $G$ be a compact Lie group, $T$ a maximal torus, ...
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0answers
38 views

Intuition of higher push-forward constant sheaves.

Let us consider the higher phsh-forward sheaves $R^if_*\mathbb{R}$ of a map $f:X\rightarrow Y$ between two compact manifolds. We assume that the fibers has a constant dimension, say $n$. I think ...
4
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0answers
31 views

Are acyclic coverings cofinal in the set of coverings?

I am interested by the following question in algebraic geometry. Recall that a covering $\mathfrak{U}$ of a topological space $X$ is acyclic for a sheaf $\mathscr{F}$ if we have $H^q(U_{i_0,\cdots, ...
2
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2answers
41 views

What is the point of triangulating topological spaces?

In a general sense, what is the purpose to triangulating, for example, a 3-dimensional topological space? What advantages does it give if we can triangulate a Seifert-Weber space into 23 tetrahedra? ...
2
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1answer
35 views

Hecke equivariance in Poincaré duality.

Consider the first singular homology and cohomology groups of a modular curve, $H^1(X,\mathbb{Z})$ and $H_1(X,\mathbb{Z})$. The Hecke algebra acts on both of them and they are dual to each other under ...
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0answers
46 views

In the Universal Coefficient Theorem, how does the cohomology generator relate to the homology generators?

Consider homology and cohomology of some space $X$ where the homology groups are finitely generated. Consider $tor(H^i(X))$, the torsion part of $H^i(X)$. How do the generators of $tor(H^i(X))$ ...
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0answers
30 views

To prove Pic$(X)$ is isomorphic to $H^1(\mathcal{U},O_X^*)$

We start with an invertible sheaf $\mathcal{L}$ and an open cover $\mathcal{U}=\{U_i\}_i$ for $X$, such that $\mathcal{L}|_{U_i}=O_X|_{U_i}$. So the line bundle is given by the information : ...
0
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2answers
91 views

Cohomology of twice-punctured plane

I am looking for pure calculus type of solution for computing the cohomology class $H^2$ of $R^2-\{p , q\}$ ie plane with two points removed. *I was not precise I apologize: Every closed 2-form is ...
3
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1answer
55 views

Why do we have that Hom is an exact functor in the situation described below?

We are given a finite $p$-group $G$ and a finite $G$-module $M$ such that $pM=0$ (therefore $M$ is in particular a $\mathbb{F}_p$-vector space). In addition we have an arbitrary $G$-module $N$ which ...
5
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0answers
55 views

There does not exist a map $S^2\times S^2\to \mathbb{CP}^2$ with odd degree.

The following is a problem from a topology qualifying exam I am studying for: Show there does not exist a map $S^2\times S^2\to \mathbb{CP}^2$ with odd degree. I think I am doing something wrong, ...
3
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0answers
49 views

For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
3
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0answers
44 views

The vanishing (?) cohomology of the Milnor fiber

Setup. Say we have a germ of a holomorphic function $f:(\mathbb C^{n+1},0)\to (\mathbb C,0)$ with a critical point at the origin. There is an $\epsilon>0$ small enough so that $f$ becomes a ...
3
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0answers
35 views

Cohomology of $S^2\times S^2/\mathbb{Z}_2$

The product of two spheres admits a diagonal $\mathbb{Z}_2$-action, $(x,y)\mapsto (-x,-y)$. I'm trying to compute the integral singular cohomology ring of the orbit space $X$ of this action. $X$ is ...
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1answer
30 views

mapping in $H_{c}^k(X,\mathbb{Z})$

If Y is closed subset of X, do anyone know what is the mapping from $H_{c}^k(X,\mathbb{Z})$ to $H_{c}^k(Y,\mathbb{Z})$ ? Is it differ if X is an open subset of Y. $H_{c}^k$ is a cohomology with ...
4
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1answer
60 views

Examples of de Rham cohomology being easier to compute that singular cohomology

De Rham's theorem states that for any smooth manifold $M$ the singular cohomology and de Rham cohomology of $M$ are isomorphic. Are there any examples of manifolds for which it is easier to compute ...
2
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1answer
92 views

De Rham cohomology, and forms on manifolds

In String Theory and M-Theory by Becker, Becker and Schwarz, they introduce a group, $$C^{p}(M)$$ which they denote the group of all closed $p$-forms on the manifold $M$. Furthermore, they state ...
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0answers
143 views

Compactly supported Dolbeault Cohomology: is this True?

nLab states that for $D$ the unit disk in $\mathbb C$, the cohomology of the complex $$ (\Omega_c^{1,\ast}(D),\overline{\partial})$$ is the continuous dual of the space of holomorphic functions ...
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1answer
31 views

Why is the generator of $H^0(S^n,\mathbb{Z})$ the identity of $H^*(S^n,\mathbb{Z})$?

I know the cohomology ring $H^*(S^n,\mathbb{Z})\simeq\mathbb{Z}[X]/(X^2)$, but the computation of this uses that fact that a generator of $H^0(S^n,\mathbb{Z})$ is the identity for the cohomology ring ...
2
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1answer
50 views

Computing $H^\bullet(\Bbb Z/n\Bbb Z)$

This is related to this other question of mine Showing that $\operatorname {Br}(\Bbb F_q)=0$ in which I also got stuck at writing a free resolution. I want to compute the group cohomology ...