The cohomology tag has no wiki summary.

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### 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 ...

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### equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space X we can define the sheaf cohomology of X in
I. the Grothendieck style (as the right derived functor of the global sections functor Gamma(X,-))
or
II. the Cech style ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### How should one think about pushforward in cohomology?

Suppose f:X→Y. If I decorate that first sentence with appropriate adjectives, then I get a pushforward map in cohomology H*(X)→H*(Y).
For example, suppose that X and Y are oriented ...

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### What is a cohomology theory (seriously)?

This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"?
I know that there exist generalized cohomology theories, Weil ...

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### “Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...

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### 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 ...

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### de Rham cohomology and flat vector bundles

I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed ...

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### Explanation for the Chern character

The Chern character is often seen as just being a convenient way to get a ring homomorphism from K-theory to (ordinary) cohomology.
The most usual definition in that case seems to just be to define ...

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### 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 ...

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**1**answer

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### Example Wanted: When Does Cech Cohomology Fail to be the same as Derived Functor Cohomology?

I want to know exactly how derived functor cohomology and Cech cohomology can fail to be the same.
I started worrying about this from this answer to an MO question, and Brian Conrad's comments to ...

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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 ...

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### What is the right version of “partitions of unity implies vanishing sheaf cohomology”

There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...

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### Explanation for the Thom-Pontryagin construction (and its generalisations)

In 1950, Pontryagin showed that the n-th framed cobordism group of smooth manifolds was equal to n-th stable homotopy group of spheres:
$$ \lim_{k \to \infty} \pi_{n+k}(S^k) \cong ...

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### Visualizing how Cech cohomology detects holes

I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space.
How can we directly visualize how and in what sense the Cech cohomology of a cover does this?
In ...

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### Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?

One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...

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### 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 ...

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### Twistings for other cohomology theories

Twistings in cohomology theories have a long history and have been used to great effect. The classical example is cohomology with local coefficients. Using this one can formulate Poincaré duality and ...

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### How to Draw Complex Line Bundles

I am giving a presentation soon on the Classification of Complex Line Bundles and I would like to have some very "basic" visualizations to use as examples.
Background and Context
I am considering ...

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### 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 ...

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### KK-theory as a stable infinity-category and KU Mod

The category KK of bivariant operator K-theory (or possibly its E-theory variant) ought to be the homotopy category of something at least close to a stable infinity-category; notably in that it ...

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### Holomorphic vector fields acting on Dolbeault cohomology

The question.
Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) ...

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### How are these algebraic and geometric notions of homotopy of maps between manifolds related?

Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and ...

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### What is classified by the (big) crystalline topos?

In his paper "Generic Galois Theory of Local Rings", G.C. Wraith states on p. 743 that the (big) crystalline topos "can be conveniently described in terms of the theory it describes". What exactly is ...

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### 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. ...

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### Is there a map of spectra implementing the Thom isomorphism?

A well known theorem in algebraic topology relates the (co)homology of the Thom space $X^\mu$ of a orientable vector bundle $\mu$ of dimension $n$ over a space $X$ to the (co)homology of $X$ itself: ...

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### What is known about K-theory and K-homology groups of (free) loop spaces?

Calculating the homology of the loop space and the free loop space is reasonably doable. There exists the Serre spectral sequence linking the homology of the loop space and the homology of the free ...

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### Cohomological characterization of CM curves

In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable ...

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### Equivalence of ordered and unordered cech cohomology.

Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:
(Ordered): ...

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### Cohomology of associative algebras

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is ...

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### Are all Galois cohomology groups also étale cohomology groups?

Let $K$ be a field and $K^s$ a separable closure of $K$, and let $\mathcal{F}$ be a sheaf on $\mathrm{Spec}(K)$ (in the étale topology).
By Grothendieck's Galois Theory, we have the isomorphism
...

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### learning crystalline cohomology

From which sources would you learn about crystalline cohomology and the de-Rham-Witt complex?

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### 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 ...

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### Intuition for Primitive Cohomology

In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\wedge \eta$. We then ...

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### Complex orientations on homotopy

I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*\mathbb{C}$P$^{\infty}$ or a statement about Thom classes. It ...

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### Cohomology of a sheaf of functions locally constant along a foliation

Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology ...

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### Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...

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### Does a “Chern character” exist for any generalized cohomology theory?

The Chern character is a ring homomorphism from the complex K-theory to the usual cohomology.
1) I wonder if there are "Chern character"-like ring homomorphisms from other generalized cohomology ...

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### Why does the Gamma function satisfy a functional equation?

In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...

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### A list of machineries for computing cohomology

This is not a question, but I just hope to hear more from everyone here on it.
A list of ready-to-use machineries to compute the de Rham / Cech cohomology of a manifold / variety. As far as I know, I ...

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### Is the following map from Z(G) x H^3(G, C*) --> H^2(G, C*) ever nontrivial?

Suppose that G is a finite group, then we have the following map f which takes an element z in the center of G and a 3-cohomology class w and returns a 2-cohomology class f(z,w) (for concreteness ...

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### Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...

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### The cohomology of a product of sheaves and a plea.

The question Consider a topological space $X$ and a family of sheaves (of abelian groups, say) $\; \mathcal F_i \;(i\in I)$ on $X$. Is it true that
$$H^*(X,\prod \limits_{i \in I} \mathcal ...

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### Group cohomology without G-modules (a.k.a. what does this bar construction compute?)

Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution.
For instance, let's take $G = \mathbb{Z}^2$, and "resolve":
$$ 0 \to ...

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### Steenrod algebra at a prime power

Let $n=p^k$ be a prime power.
When $k=1$, the algebra of stable operations in mod $p$ cohomology is the Steenrod algebra $\mathcal{A}_p$. It has a nice description in terms of generators and ...

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### How to get product on cohomology using the K(G, n)?

This came up in the question about Eilenberg-MacLane spaces. Given the definition of K(G, n), it's easy to prove that there is a map ...

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### What kind of geometric operations “scale up” cohomology?

There's an obvious operation on the category of graded rings, given by "scaling up," multiplying the grading of every element by some fixed constant.
Does anyone know of an operation on the level of ...