This tag is for questions relating to cohomology groups and cochain complexes.
56
votes
2answers
2k views
Direct proof that the wedge product preserves integral cohomology classes?
Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$.
There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...
34
votes
2answers
810 views
Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres
So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
16
votes
3answers
791 views
Motivating Cohomology
Question: Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?
Why ...
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 ...
14
votes
1answer
385 views
Group cohomology versus deRham cohomology with twisted coefficients
Let $G$ be a simple simply-connected Lie group, let $M$ be a 3-manifold and $P \to M$ a principal $G$-bundle. Let $A$ be a flat connection in this bundle, and let $\text{Ad} P$ be the associated ...
13
votes
1answer
734 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?
11
votes
1answer
232 views
What functor does $K(G, 1)$ represent for nonabelian $G$?
For $G$ an abelian group, the Eilenberg-Maclane space $K(G, n)$ represents singular cohomology $H^n(-; G)$ with coefficients in $G$ on the homotopy category of CW-complexes. If $n > 1$, then $G$ ...
11
votes
2answers
135 views
Different ways of representing a second cohomology class
There are probably many ways of talking about a second (integral) cohomology class of a smooth, closed, orientable manifold $M$ of dimension $n$. Here are a few, with $\alpha\in H^2(M,\mathbb{Z})$:
...
11
votes
1answer
331 views
Why isn't $H^*(\mathbb{R}P^\infty,\mathbb{F}_2)\cong \mathbb{F}_2[[x]]$?
We just computed in class a few days ago that $$H^*(\mathbb{R}P^n,\mathbb{F}_2)\cong\mathbb{F}_2[x]/(x^{n+1}),$$ and it was mentioned that $H^*(\mathbb{R}P^\infty,\mathbb{F}_2)\cong \mathbb{F}_2[x]$, ...
10
votes
2answers
463 views
Meaning of “efface” in “effaceable functor” and “injective effacement”
I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...
10
votes
2answers
370 views
Which cohomology theories have a formula $\langle \Omega,\text d \omega \rangle = \langle \partial \Omega,\omega \rangle$?
Is a formula
$$\langle \Omega,\text d \omega \rangle = \langle \partial \Omega,\omega \rangle$$
like Stokes theorem
$$\int_\Omega \text d \omega=\int_{\partial\Omega} \omega$$
common in cohomology ...
10
votes
1answer
584 views
Topological vs. Algebraic $K$-Theory
Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras ...
10
votes
2answers
305 views
Does every Poisson bracket on a commutative algebra come from a second-order deformation?
Let $A$ be a commutative algebra over a field $k$ (of characteristic not equal to $2$ to be safe). Recall that $f : A \otimes A \to A$ is a Hochschild $2$-cocycle if it satisfies
$$f(ab, c) + f(a, b) ...
9
votes
1answer
113 views
Proof of $H^k(X,\mathbf k) = H^k(X,\mathbb Z) \otimes \mathbf k$
Let $X$ be a compact manifold and denote $H^k(X,G)$ the $k$-th cohomology group with coefficients in the abelian group $G$.
Using Cech cohomology one can prove that there is a natural isomorphism $ ...
9
votes
1answer
289 views
vector bundles on affine schemes
Serre's theorem (one of them) states that for a quasi-coherent sheaf $\mathscr F$ on an affine noetherian scheme $H^i(X,\mathscr{F})$ vanish for $i >0$. I used to think that this would imply that ...
9
votes
1answer
274 views
Étale cohomology of projective space
I have some very basic question about étale cohomology.
Namely I would like to compute the étale cohomology of of the projective space over the algebraic closure of $\mathbb F_q$ along with its ...
8
votes
3answers
614 views
How to define Homology Functor in an arbitrary Abelian Category?
In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient
Ker d / Im d
where d as usual denotes the differentials, indexes skipped for simplicity.
How ...
8
votes
3answers
457 views
The “need” for cohomology theories
In many surveys or introductions, one can see sentences such as "there was a need for this type of cohomology" or "X succeeded in inventing the cohomology of...".
My question is: why is there a need ...
8
votes
3answers
222 views
Turning cobordism into a cohomology theory
I've recently finished one semester in differential topology (with Milnor's Topology from the Differentiable Viewpoint) and my first semester of algebraic topology. I believe I understand Milnor's ...
8
votes
3answers
607 views
Toy sheaf cohomology computation
I asked this question a while back on MO : http://mathoverflow.net/questions/32689/how-should-a-homotopy-theorist-think-about-sheaf-cohomology
One thing that really helped in learning the Serre SS ...
8
votes
2answers
444 views
Calculating the cohomology with compact support of the open Möbius strip
I am having problems calculating the cohomology with compact support of the open Möbius strip (without the bounding edge).
I am using the Mayer Vietoris sequence: U and V are two open subsets ...
8
votes
1answer
224 views
Finite groups with periodic cohomology
I'm trying to understand Chapter 12, Section 11 in Cartan + Eilenberg's Homological Algebra, which concerns finite groups with periodic cohomology. Unfortunately I am jumping right to this section in ...
8
votes
0answers
204 views
Why do universal $\delta$-functors annihilate injectives?
Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to ...
7
votes
4answers
604 views
Poincare Duality Reference
In Hatcher's "Algebraic Topology" in the Poincare Duality section he introduces the subject by doing orientable surfaces. He shows that there is a dual cell structure to each cell structure and it's ...
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, ...
7
votes
2answers
123 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 ...
7
votes
2answers
90 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?
7
votes
2answers
237 views
Manifold with 3 nondegenerate critical points
Suppose $M$ is a n-dimensional (compact) manifold and $f$ is a differentiable function with exactly three (non-degenerate) critical points. Then one can show, using Morse theory, that $M$ is ...
7
votes
1answer
173 views
Confusion on Cech cohomology
From Harvard math qualification exam, 1990.
Let $X$ be a smooth manifold with an open cover $N<\infty$ sets $\{B_{n}\}^{N}_{1}$ which are contractible. Assume that $$\pi_{0}(B_{n}\cap B_{m})\le ...
7
votes
2answers
133 views
Geometric interpretation of injective/projective resolutions?
I understand the geometric interpretation of derived functors, as well as their usefulness in giving a simple, purely algebraic description of cohomology.
I also understand how resolutions are used ...
7
votes
1answer
430 views
Homology of the Empty set
I am under the impression that the standard convention for the homology (singular) of the empty set is 0 in all non negative degrees and $\mathbb{Z}$ in degree $-1$. I have no problem with this ...
7
votes
1answer
243 views
How calculate the De Rham cohomology group of $3$-torus: $T^3$?
How do I calculate the De Rham cohomology group of the $3$-torus $T^3$? Here $T^3=S^1 \times S^1 \times S^1 $.
Using the Mayer-Vietoris sequence, I can show that $\dim H_3(T^3)=\dim H_0(T^3)=1$. But ...
7
votes
1answer
86 views
If $H_n(X;\mathbb{Z})$ are all f. g. free abelian, then $H^*(X;\mathbb{Z}) \otimes \mathbb{Z}_p \cong H^*(X; \mathbb{Z}_p)$?
An exercise in Hatcher's book asks to prove that whenever $X$ is a space with the homology groups $H_n(X; \mathbb{Z})$ finitely generated free abelian for each $n \geq 0$, then $H^*(X; \mathbb{Z}) ...
6
votes
3answers
498 views
Cohomology easier to compute (algebraic examples)
There is a previous post about motivating cohomology and it contains much of differential geometry examples, something I have just started and still have to figure out. It is said that one uses ...
6
votes
4answers
386 views
What do higher cohomologies mean concretely (in various cohomology theories)?
Superficially I think I understand the definitions of several cohomologies:
(1) de Rham cohomology on smooth manifolds (I understand this can be probably extended to algebraic settings, but I haven't ...
6
votes
2answers
243 views
No torsion in $H^1_c(X,\mathbf{Z})$?
If $X$ is a very nice topological space, for example a finite simplicial complex, then is it true that the cohomology with compact supports $H^1_c(X,\mathbf{Z})$ is torsion-free? I have seen an ...
6
votes
1answer
208 views
What tells rational cohomology about integral cohomology?
Say we have a finite CW complex with cells only in even degrees. For example a $\mathbb {CP}^n$ or a complex flag variety. If we know the rational cohomology ring, does it also determine the integral ...
6
votes
1answer
192 views
Is there any relation about rational homology of X and X/G
If we know the rational homology of X is 0, can we get some information about the rational homology of X/G, where G is a finite group? Thank you very much for the answers!
6
votes
2answers
238 views
Cohomology ring $H^*(\mathbb{R}P^3 \# \mathbb{R}P^3; \mathbb{Z}_2)$.
I am interested in computing the cohomology ring $H^*(\mathbb{R}P^3 \# \mathbb{R}P^3; \mathbb{Z}_2)$. Here # is the connected sum. Using a suggestion here on my earlier post, I computed the additive ...
6
votes
1answer
103 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 $Hom(F^{\cdot},-)$. ...
6
votes
1answer
129 views
How does Pontryagin duality fit into the general cohomology theory framework?
Pontryagin duality implies the isomorphic relation of the function space $C(G)$ on a locally compact group $G$ to the function space on it's dual group $\hat G \overset{\sim}{=}\text{Hom}(G,T)$, ...
6
votes
1answer
258 views
Group cohomology and Shapiro's lemma
This is a stupid question about group cohomology, but it confuses me a lot. Basically I think that the problem is the fact that I do not really understand Shapiro's lemma.
Say we take a profinite ...
6
votes
0answers
78 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 ...
6
votes
0answers
81 views
Computing a specific direct limit
Suppose we have the following sequence of $\mathbb{Z}$-modules
$$G\,\,\overset{M}{\longrightarrow}\,\, G\,\,\overset{M}{\longrightarrow}\,\, G\,\,\overset{M}{\longrightarrow}\,\, \cdots,$$
where each ...
5
votes
1answer
268 views
cohomology vs homology
I have learned the basic things about cohomology and homology. It seems that homology and cohomology both deal with the same objects, the complexes, but with a different choice of the indexes (for ...
5
votes
2answers
209 views
About the definition of Cech Cohomology
Let $X$ be a topological space with and open cover $\{U_i\}$ and let $\mathcal F$ be a sheaf of abelian groups on $X$. A $n$-cochain is a section $f_{i_0,\ldots,i_n}\in U_{i_0,\ldots,i_n}:= ...
5
votes
1answer
133 views
Geometric invariants of a scheme
Following my previous question about sheaf cohomology, I'd like to ask about its applications to algebraic geometry. I have now learned a little about homological algebra and I can see that for ...
5
votes
1answer
114 views
Are maps inducing the same cohomology homomorphisms homotopic?
It is not hard to show that given $f,g: X \rightarrow Y$, with $f$ and $g$ homotopic the induced homomorphisms $f^*, g^* : H^* (Y, \mathbb{Z}) \rightarrow H^* (X, \mathbb{Z})$ are the same.
Is the ...
5
votes
1answer
224 views
Torsion-free virtually-Z is Z
It is well known that a torsion-free group which is virtually free must be free, by works of Serre, Stallings, Swan...
Is there a simple cohomological proof of the fact that a torsion-free group ...
5
votes
1answer
282 views
Varying definitions of cohomology
So I know that given a chain complex we can define the $d$-th cohomology by taking $\ker{d}/\mathrm{im}_{d+1}$. But I don't know how this corresponds to the idea of holes in topological spaces (maybe ...
