The cohomology tag has no usage guidance.

**28**

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**10**answers

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### What is (co)homology, and how does a beginner gain intuition about it?

This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...

**64**

votes

**2**answers

14k views

### Why is the Hodge Conjecture so important?

The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm ...

**45**

votes

**2**answers

4k views

### Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...

**29**

votes

**4**answers

2k views

### Integral cohomology (stable) operations

There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its ...

**11**

votes

**6**answers

3k views

### Cohomology of fibrations over the circle

Are there any general results on the (integral) cohomology of manifolds that are fibrations over the circle? Any literature references much appreciated.

**1**

vote

**1**answer

210 views

### permutation action on cohomology of Stiefel manifolds

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds.
In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ...

**17**

votes

**1**answer

1k views

### For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?

My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...

**-1**

votes

**1**answer

175 views

### A closed manifold with a subset with the same ring cohomology

Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies?
In this question ...

**57**

votes

**5**answers

8k views

### What is sheaf cohomology intuitively?

What is sheaf cohomology intuitively?
For local systems it is ordinary cohomology with twisted coefficients. But what
if the sheaf in question is far from being constant?
Can one still understand ...

**44**

votes

**11**answers

6k views

### Why torsion is important in (co)homology ?

I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...

**31**

votes

**2**answers

2k views

### Why should I care about topological modular forms?

There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...

**14**

votes

**2**answers

2k views

### BRST cohomology

I am reading some work on Mirror Symmetry from Physics perspective,the physicists seem to use some aspects of BRST quantization and BRST cohomology. What is BRST Quantization and BRST cohomology, in ...

**45**

votes

**7**answers

3k views

### Can anyone give me a good example of two interestingly different ordinary cohomology theories?

An answer to the following question would clarify my understanding of what a cohomology theory is. I know it's something that satisfies the Eilenberg-Steenrod axioms, and I know that those axioms ...

**22**

votes

**4**answers

3k views

### de Rham vs Dolbeault Cohomology

For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. ...

**20**

votes

**0**answers

633 views

### Calabi-Yau cohomology?

My question here is going to be this -- but I'll give a bit of background to explain myself in a moment:
What has been done/what results are available on Calabi-Yau cohomology in degree $n \geq 3$ ...

**18**

votes

**1**answer

1k views

### De Rham cohomology of formal groups

Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F_w(X)$ be ...

**7**

votes

**3**answers

380 views

### Are generalized cohomology theories a set of complete homotopy invariants for spaces ?

In the same vein of this MO question, one can ask:
If two spaces $X$, $Y$ have isomorphic generalized cohomology rings $\mathrm{h}^{\bullet}(X)\cong \mathrm{h}^{\bullet}(Y)$ for every ...

**6**

votes

**4**answers

980 views

### Cohomology groups of homogeneous spaces

Is there a general method to calculate the cohomology groups of homogeneous spaces ($G/H$), such as $\frac{U(4)}{U(2)\times U(2)}$, $\frac{U(5)}{U(2)\times U(3)}$, $U(4)/U(2)$, etc. If yes, could you ...

**25**

votes

**1**answer

741 views

### Making $\mathbb{Q}$-cohomology integral

Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$
with $i>0$.
Does there always exist a variety $Y$ and a ...

**15**

votes

**2**answers

903 views

### Are the homology and cohomology Serre spectral sequences dual to each other?

If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...

**9**

votes

**2**answers

520 views

### “Skew Cohomology” of a Space

Let $X$ be a space. The symmetric group $\Sigma_{n+1}$ acts on the function space
$$
X^{\Delta^n}
$$
of continuous maps from the standard $n$-simplex to $X$. The action is induced
by permuting the ...

**7**

votes

**3**answers

588 views

### Relation between cohomology of ordered and unordered configuration spaces?

For any manifold $M$, the unordered configuration space of $k$ points is obtained as a quotient of ordered configuration space of $k$ points by the group action of symmetric group on $k$ letters. Does ...

**5**

votes

**3**answers

957 views

### Brauer group of projective space

I've read that $\text{Br} \mathbb{P}^n_k$ (here $\text{Br}$ is the cohomological Brauer group, i.e. $H^2_{ét}(-,\mathbb{G}_m)$) is just isomorphic to $\text{Br} k$. As proof of this fact seems to be ...

**4**

votes

**1**answer

390 views

### Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...

**10**

votes

**1**answer

469 views

### positions of a methane molecule with carbon atom at the origin

Let $\text{CH}_4$ be the molecule of Methane:
The four hydrogen atoms form vertices of a regular tetrahedron with the carbon atom in the center of the regular tetrahedron.
Here we regard all atoms ...

**8**

votes

**4**answers

644 views

### $Sq^1$ cohomology of spaces

For any space $X$, the first Steenrod square cohomology operation
$$Sq^1\colon H^\ast(X;\mathbb{Z}_2)\to H^{\ast +1}(X;\mathbb{Z}_2)$$
is a derivation, meaning that $Sq^1\circ Sq^1 = 0$ and ...

**7**

votes

**1**answer

397 views

### Intuitive Aproach to Dolbeault Cohomology [closed]

(Duplicated from math.stackexchange)
I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. ...

**7**

votes

**3**answers

365 views

### Group Extensions and Line Bundles on $BG$

I am sure the answer to this question is well-known, but
It is well known that the group cohomology $H^2(G,\mathbb Z)$ classifies group extensions $0\to \mathbb Z\to E\to G\to 1$ and that for a ...

**3**

votes

**0**answers

1k views

### group cohomology and cohomology of classifying space [closed]

Let $G$ be a discrete group, and $BG$ is the classifying space.
It is well-known that the group cohomology of $G$-module M, is the same as the cohomology on $BG$ with coefficient in $\tilde{M}$, which ...

**5**

votes

**2**answers

295 views

### computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit

Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say ...

**5**

votes

**1**answer

816 views

### Natural morphism appearing in Grothendieck spectral sequence

Assume we are in the setting of the Grothendieck spectral sequence (Weibel, 5.8): $G : A \to B, F : B \to C$ are left exact functors such that $G$ sends injective objects to $F$-acyclic objects. Now ...

**4**

votes

**2**answers

859 views

### Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has
$$\otimes: D^{-}(X) \times D^{-}(X) \to ...

**4**

votes

**1**answer

713 views

### The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base

Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. ...

**2**

votes

**1**answer

364 views

### vanishing theorems

I would be glad to know about possible generalizations of the following results:
1) (Grothendieck) Let $X$ be a noetherian topological space of dimension $n$. Then for all $i>n$ and all sheaves of ...

**8**

votes

**1**answer

450 views

### Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function?

Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta ...

**7**

votes

**1**answer

430 views

### Do modular forms show up in the cohomology of moduli spaces of unmarked curves?

Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + ...

**6**

votes

**2**answers

168 views

### An example of two cofibrant dg categories whose tensor product is not cofibrant

I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and ...

**3**

votes

**0**answers

157 views

### Brauer group of a rational variety

This is a follow-up question to this question. There and here $X$ is a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. My question is:
...

**1**

vote

**2**answers

382 views

### integral or rational cohomology of real grassmannians

I have obtained that the cohomology rings
$$
H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k].
$$
Also
$$
H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...

**8**

votes

**2**answers

455 views

### Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$

Let $X$ be a nice topological space and denote by $\pi_1(X)$ its fundamental group.
It is well-known that there is a well-defined map
$$
0 \rightarrow H^2(\pi_1(X),A) \rightarrow H^2(X,A),$$
where ...

**6**

votes

**2**answers

447 views

### Hard Lefschetz Theorem for the Flag Manifolds

In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See herehere for a description.) I would like to know what the hard Lefschetz ...

**4**

votes

**0**answers

163 views

### Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...

**4**

votes

**1**answer

258 views

### Künneth formula for Bredon cohomology theory

Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...

**3**

votes

**2**answers

249 views

### cohomology algebra of braid spaces, configuration spaces

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for ...

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votes

**0**answers

367 views

### Differential and pre-differential of Jacobi identity

Let M be a manifold.
To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied?
That is a Lie algebra structure for which $[X,fY]=f[X,Y]$.
(For ...

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vote

**4**answers

331 views

### Global section of very ample line bundles and its value on stalks

Let $X$ be a projective scheme and $\mathcal{L}$ be a very ample line bundle on $X$ with respect to some projective embedding $X \hookrightarrow \mathbb{P}^n_{\mathbb{C}}$ (for some $n$).
Given any ...

**1**

vote

**1**answer

95 views

### A cohomology associated with a codimension one foliation

Let $\alpha$ be a non vanishing one form on a manifold which which defines a codimension one foliation. With this $\alpha$ we define the following complex:
$$\phi:\Omega^{i}(M)\to ...

**1**

vote

**2**answers

237 views

### A cohomology associated to a 1- form

In this question all objects are real analytic.(manifolds, differential forms..)
Assume that $M$ is a compact manifold and $\alpha \in \Omega^{1}(M)$ is a one form.
We define a map ...

**1**

vote

**1**answer

313 views

### $\pi$-cohomology class — a variant of cohomology class

Let $X$ be a topological space with a triangulation. The triangulation defines a
chain complex in $X$. Let $\mu_d$ be a cochain and $M^d$ be a chain. We use $<
\mu_d, M^d > \in M$ to denote ...