This tag is for questions relating to cohomology groups and cochain complexes.
0
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0answers
22 views
Is the intersection of compact Stein sets, a compact Stein set?
Brian Conrad, in an article of his, defined a compact Stein set to be a compact set (subset of a complex manifold) K admitting Hausdorff neighborhood such that H^i(K,G)=0 for all i>0, and for all G ...
4
votes
2answers
69 views
How do I know when a form represents an integral cohomology class?
Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class.
I would like to ...
2
votes
3answers
72 views
What does $p+q=k$ mean in the index of summation?
I need help solving something I don't understand. OK so the problem is this:
$$H^k(X,C)=\bigoplus_{p+q=k} H^{p,q}(X),$$
What does the $\;p+q=k\;$ mean?
Thank you anybody that helps! :)
1
vote
0answers
19 views
Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$
What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
1
vote
0answers
17 views
Chern Character Isomorphism for non-CW complexes
Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that $\mathrm{K}^\ast(X) \otimes \mathbb{Q} \cong \mathrm{H}^\ast(X; \mathbb{Q})$, where $\mathrm{H}^\ast$ denotes ...
1
vote
0answers
46 views
Explanation of notations
I was reading Binary icosahedral group in Wikipedia. The author uses $<2,3,5>$ and $(2,3,5)$ to denote the groups. And the Coxeter group of type $H_4$ is denoted by $[3,3,5]$. Could anyone ...
0
votes
0answers
29 views
K and KO spectra
In Switzer's algebraic topology book, ch 11 page 216, he defines the K and KO spectra. He then goes on to say: "Since they are $\Omega$-spectra, we have
$\tilde{KO}^0(X) \cong [X,x_0;\mathbb{Z} ...
1
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0answers
30 views
Dolbeault cohomology of $S^{2n-1} \times S^1$
Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}}(X)$ and $H^{(0,1)}_{\bar{\partial}}(X)$. I don't know how to do this but if we use Kunnet formula we have that ...
1
vote
1answer
57 views
filtration on the (co)homology of a space from the filtration of a space
Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let
...
0
votes
1answer
23 views
Prove that the $2$ form defines a symplectic structure
Prove that the $2$ form
$$\omega = -2[(1+x_2^2)dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_3 \wedge dx_4]$$
defines a symplectic structure on $\mathbb{R}_x^4$.
My definition of as ...
3
votes
1answer
45 views
Sufficient condition for a direct limit of abelian groups to be infinitely generated
I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
7
votes
5answers
103 views
(Elementary) applications of group (co-)homology
I am looking for an elementary example of a problem, for which one does not need many things to understand the question, but which can be solved with group homology or cohomology.
My background is, ...
1
vote
0answers
43 views
Cohomology and 1-forms with compact support
I'm, having troubles with the following
Let $U$ be a bounded open set in $\mathbb{R}^{2}$ such that $\mathbb{R}^{2}\setminus U$ has $n+1$ connected components. Prove that $\dim(H_c^{1}(U))=n$.
I ...
6
votes
0answers
79 views
When does a cohomology theory have a ring structure?
I've looked around and I can't quite seem to find an answer to this question. When does a cohomology theory admit a non trivial product structure? I was trying to compute a cohomology ring from a CW ...
3
votes
1answer
30 views
Cohomology theories that arise in different fields of mathematics
During my studies in university I have encountered several cohomology theories. Part of them I've met in topology\differential geometry\analysis on manifolds courses (simplicial, singular, cell, ...
5
votes
0answers
51 views
Injective Resolutions in $\mathfrak{Ab}(X)$
Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, ...
7
votes
2answers
126 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 ...
4
votes
1answer
39 views
Cohomology of $\Bbb CP^{\infty}=BU_1, BU_2,\dots$ : A reference request
Where can I find the calculation of the cohomology rings of the classifying spaces $BU_n,~BO_n$ and $BO,~BU$? I took a class where extensive use was made of these cohomology rings, but I missed the ...
3
votes
0answers
31 views
Intuitive definition of Čech cohomology for compact surfaces
Let $X$ be a smooth compact $k$-surface in $\mathbb R^n$ without boundary. Today on my lection lecturer introduced Čech cohomology as follows (not like in Wikipedia): let $\mathcal U$ be a finite open ...
5
votes
0answers
34 views
“Inverse problem” for Brauer groups
This question is really just a curiosity, but I'm really interested in the answer.
Given a field $K$, we can form the set$^*$ $Br(K)$ consisting of equivalence classes of finite-dimensional central ...
2
votes
1answer
34 views
Why functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$ are called cocycles?
Let $X$ be some smooth manifold and $\{U_\alpha\}$ be its open cover. The last month I hear very often that one calls a collection of functions $f_{\alpha \beta} \colon U_\alpha \cap U_\beta \to Y$, ...
0
votes
1answer
47 views
Restriction map on a compact orientable manifold without a boundary.
I have the following problem:
Let $M$ be and $n$-dimensional compact oriented manifold without boundary. Let $p\in M$ be and point and let $M_p=M\backslash\{p\}$. Let $j:S^{n-1}\to M_p$ be the ...
1
vote
0answers
40 views
Cohomologies of $\mathbb R^n$ with rational differential forms
We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
3
votes
1answer
76 views
Order of the first cohomology group and subgroups
Let $M$ be a $G$-module and $H$ a subgroup of $G$. Is $\# H^{1}(H, M) < \# H^{1}(G, M)$?
2
votes
2answers
70 views
Equivalence of categories and derived functors.
Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the ...
2
votes
0answers
39 views
Simplicial cup product on torus
I'm trying to compute the simplicial cup product on the torus (using $\Delta$-complexes) but running into a problem: each way I draw the fundamental polygon I get different answers! When I draw it as ...
2
votes
1answer
50 views
Simplicial cohomology of $ \Bbb{R}\text{P}^2$
I've managed to confuse myself on a simple cohomology calculation. I'm working with the usual $\Delta$-complex on $X = \mathbf{R}\mathbf{P}^2$ and I've computed the complex as ...
3
votes
0answers
84 views
Simple exercise in cohomology
I know this is a simple exercise but I am stuck unfortunately.
Question:
Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
3
votes
1answer
85 views
Algebraic Topology Double Complexes
I am going through Bott and Tu and trying to do Exercise 9.13 which says
When a homomorphism $f: K \rightarrow K'$ of double complexes induces $H_d$-isomorphism, it also induces $H_D$-isomorphism.
...
2
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0answers
57 views
Prove Poincare duality theorem with Morse theory.
First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
2
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0answers
42 views
$f^\ast (a \smile b) = f^\ast(a) \smile f^\ast(b)$ using simplicial chains to define cochains
Let $f \colon X \to Y$ be a continuous map between topological spaces $X$ and $Y$, $f_\ast$ be the induced homomorphism of singular chains $C_k^s(X;G)$, $C_k^s(Y;G)$ and $f^\ast$ be the induced ...
0
votes
0answers
43 views
De Rham Cohomology of Product of Manifold with an Open Interval
Let $X$ be a submanifold of $\mathbb{R}.$ Prove that $H^{k}_{DR} (X) = H^{k}_{DR} (X\times (0,1)).$ I know that we should consider maps $\iota_a: X\to X\times (0,1)$ by $\iota_a(x) = (x,a)$ for ...
2
votes
1answer
94 views
A table of homology and cohomology groups
Does anyone know where I can find a table of the homology and cohomology groups, with different coefficients, of standard spaces - $S^1\times S^1$, Klein bottle, projective space, etc.?
1
vote
0answers
45 views
Cohomology of a chain complex
I know that one can define a chain complex for a CW complex X by taking the chain groups $C_n(X)$ as the free group generated by the $n$-cells, $C_n(X;\mathbb{Z}) = \mathbb{Z}\langle ...
1
vote
0answers
17 views
Sending the Poincaré dual class of a point in a smooth manifold to 1
I have been given the following problem:
For a compact oriented $n$-dimensional manifold, use a nowhere zero $n$-fold $\omega\in\Omega^n(M)$ to define a linear map
\begin{equation}
[M]: H^n(M)\to ...
3
votes
1answer
55 views
What is the cohomological explanation for the Condorcets voting paradox?
according to the nlab entry on the Condorcet Paradox in social choice (that is voting preferences may be circular even if voters preferences are not) has a cohomological explanation - what is it?
2
votes
0answers
70 views
Vanishing of local cohomology of constructible sheaves
Recall, that if $\mathcal{F}$ is a coherent sheaf on a variety and $Z$ is an l.c.i. subvarity of codimension $n$, then $H^i_Z(\mathcal F)$ vanishes for $i > n$.
Is there an analogous statement for ...
1
vote
0answers
25 views
Relation between the pull back and the pull forward map on the (co)homology groups
So let $f:X\rightarrow Y$ be a continuous map. Is there any relation between
$$f_{*}:H_{k}(X)\rightarrow H_{k}(Y)$$
and
$$f^{*}:H^{k}(Y)\rightarrow H^{k}(X).$$
7
votes
2answers
91 views
Dimension of de Rham Cohomology groups?
Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
1
vote
1answer
55 views
Induced sequence of global sections
I'm reading Differential Analysis on Complex Manifolds by Raymond O. Wells. It states the following in the beginning of section 3 of chapter 2 on page 51:
Consider a short exact sequence of sheaves:
...
6
votes
1answer
104 views
How to compute Hom in derived category?
Let $X$ be a smooth variety, $D^{b}(X)$ be the derived category of bounded coherent sheaves.Then there is a definition of $Hom(F^{\cdot},G^{\cdot})$ which is the derived functor of $Hom(F^{\cdot},-)$. ...
5
votes
1answer
95 views
When is a map essential in Čech cohomology?
I read a nice survey of parts of game theory, Foundations of Strategic Equilibrium, by Hillas and Kohlberg. Something where I stumble is the discussion of Mertens stability. There is a definition that ...
4
votes
0answers
55 views
Example where Čech and derived functor cohomologies don't agree.
I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces.
Is there a simple example of a topological space ...
1
vote
0answers
61 views
De Rham cohomology of $\mathbb R^3$ without lines and a circumference
I am trying to calculate De Rham cohomology of the following spaces:
$X=\mathbb R^3\setminus r$ where $r$ is a line;
$Y=\mathbb R^3\setminus (r \cup \gamma)$ where $r$ is a line and $\gamma$ is a ...
15
votes
1answer
236 views
Twisted Cech cohomology
Let $X$ be a CW-complex with contractible universal cover $\tilde{X}$ and fundamental group $\pi = \pi_1X$. Twisted (co)homology is found by lifting the cell structure on $X$ to a $\pi$-invariant ...
1
vote
1answer
16 views
What is the map $\pi^*:H^{\dim X/G}(X/G,Z)\rightarrow H^{\dim X}(X,Z)$?
Let $\pi:X\rightarrow X/G$ be a free quotient map by a finite group $G$. Assume that both $X$ and $X/G$ are oriented. We know that $\pi_*$ maps the fundamental class $[X]$ to $|G|[X/G]$.
What about ...
2
votes
1answer
65 views
Cohomology $SO(3)$
We have that De Rham cohomology of $SO(3) \simeq \mathbb{R}P^{3}$ is $\mathbb{R}$ in degree $0$ and $3$ and $0$ in degree $1$ and $2$. But I saw that $H^{*}(SO(3)) \simeq \mathbb{Z}_{2} $ in degree 2. ...
0
votes
1answer
30 views
Cellular cohomology complex projective spaces
I have to calculate the cohomology of complex projective spaces $\mathbb{C}P^{n}$ using cellular cohomology (I know that we have a CW decomposition of $\mathbb{C}P^{n}$ in $n+1$ cells of even ...
0
votes
0answers
43 views
Cohomology of Stiefel manifold
How can I compute $H^{*}(V_{k}(\mathbb{C}^{n}))$? Where I denote with $V_{k}(\mathbb{C}^{n})$ the Stiefel manifold of $k$-frame in $\mathbb{C}^{n}$.
0
votes
0answers
35 views
Chern classes fiber bundle projective space
I calculated the cohomology of $\mathbb{C}P^{n}$ using the spectral sequence associate to the fibration $S^{1} \hookrightarrow S^{2n+1} \rightarrow \mathbb{C}P^{n}$. How can I find the first Chern ...
