**56**

votes

**8**answers

5k views

### 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 Čech ...

**1**

vote

**1**answer

255 views

### cohomology of the orbit space of a group action

Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain ...

**6**

votes

**0**answers

107 views

### What is the etale cohomology of a product?

Let $X$ be a (smooth, projective if you wish) variety over a field $k$. I'm mostly interested in $k=\mathbb{F}_q$.
What can one say about the etale cohomology ring $H_{et}^*(X \times_k X)$, say for ...

**5**

votes

**1**answer

289 views

### Known results in the Cohomology of finite groups

I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...

**31**

votes

**10**answers

5k views

### 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 ...

**14**

votes

**3**answers

3k views

### 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 ...

**3**

votes

**0**answers

114 views

### What is the integral cohomology of an Enriques surface over a finite field?

Probably this is well-known, but I could not find it. I would like to understand the integral $2$-adic etale cohomology of an Enriques surface over $\mathbb{F}_q$ in dimension 2: $H_{et}^2(X, ...

**6**

votes

**1**answer

176 views

### Cohomology ring of a fiberwise join

I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...

**16**

votes

**2**answers

355 views

### $G$-action on the integral homology of a compact surface

Let $S$ be a compact connected orientable surface, and let $G$ be a nontrivial finite group acting freely on $S$ and preserving orientation (note the the action being free is a strong condition, since ...

**4**

votes

**2**answers

194 views

### Equivalence of different cohomology groups

Let $X$ be a topological space (may be assumed to be locally compact). Let $A$ be either a field or $\mathbb{Z}$. One can consider various cohomology groups:
(1) singular cohomology ...

**15**

votes

**2**answers

349 views

### Table of (integral) cohomology groups of K(Z,n)

Can I find somewhere a table of the (first few) cohomology groups of $K(\mathbb{Z},n)$ with integer coefficients?
It seems like a natural counterpart to the table of the homotopy groups of spheres, ...

**2**

votes

**2**answers

209 views

### Cohomology of Homogeneous Complex Manifolds

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...

**3**

votes

**0**answers

356 views

### Finding the number of rational points effectively

Consider $\# P$ and $\oplus P$.
There is a $\# P$-hard problem: to find number of rational solutions of a system of polynomial equations over $\mathbb{F}_2$. The corresponding $\oplus P$-hard ...

**5**

votes

**1**answer

205 views

### Group cohomology of the cyclic group

It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution,
$\require{AMScd}$
\begin{CD}
\cdots @>N>> \mathbb ...

**4**

votes

**0**answers

62 views

### Relation of BRST model of equivariant cohomology and BRST cohomology?

I'm now reading Kalkman's paper "BRST model for equivariant cohomology and representation for equivariant Thom class". And I've seen his definition for BRST model is
$B=W(\mathfrak{g})\otimes ...

**4**

votes

**0**answers

49 views

### Reference request: local cohomology in disjoint union

I have a topological space $X$ and two disjoint, closed subspaces $Y$ and $Z$ of $X$. I believe that in this situation, for any abelian sheaf $\mathcal{F}$ on $X$ and any $p \in \mathbb{N}$, there is ...

**5**

votes

**0**answers

121 views

### Two natural maps asssociated with the nerve of a cover

Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...

**11**

votes

**0**answers

190 views

### Cohomology of a configuration space of points on $\mathbb C^\times$ with an additional restriction

Let $Conf_{1,n}^3$ be the configuration space of collections of $n$ distinct numbered points on the annulus $\mathbb C^\times$ with an imposed restriction: for any $r\in \mathbb R^+$ the circle ...

**2**

votes

**1**answer

167 views

### Third (co-) homology of Cyclic groups

Is there a general simple theorem for the third cohomology of cyclic groups $H_3(\mathbb{Z}_n, U(1))= ?$. In particular, I am interested in finding $H_3(\mathbb{Z}_8, U(1))$. I know the answer can be ...

**11**

votes

**2**answers

3k views

### Understand Cech Cohomology

I am currently trying to understand Cech cohomology. Five questions arised and I would be glad for help. In what follows $X$ is a topological space.
I really like Dugger's and Isaksen's paper ...

**10**

votes

**1**answer

383 views

### Differential geometric interpretation of cohomology

I'm not sure whether this is an appropriate question for this forum. I'm afraid that this is not a research level question however:
1. It's about reference request therefore the answer does not ...

**14**

votes

**8**answers

1k views

### 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 ...

**4**

votes

**1**answer

219 views

### For a finite flat (etale?) morphism $f:Y\to X$, is $f_*1_Y-\deg f . 1_X$ nilpotent in $A^0(X)$, where $A^*$ is the algebraic cobordism?

Let $A^\ast$ be an algebraic oriented cohomology theory (i.e. it is equipped with certain push-forwards for projective morphisms of smooth varieties over the base field $k$; see section 2 of ...

**2**

votes

**0**answers

71 views

### why is $\cap \mu_B:H^k(\mathbb{R}^n,\mathbb{R}^n\setminus B;R)\to H_{n-k}(\mathbb{R}^n;R)$ an isomorphism? [closed]

I asked this http://math.stackexchange.com/q/1694046/309968 question already on MSE, but received no answer and I hope it's ok if I ask here for once.
Let $R$ be commutative ring with $1_R$
Lemma: ...

**1**

vote

**0**answers

78 views

### Reference: Relative cohomology of a morphism

Let $f\colon Y \to X$ be a morphism of schemes, the inverse image in $K$-theory always fit into a long exact sequence
$$
\cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots
$$
where the ...

**8**

votes

**2**answers

882 views

### A simple proof of the Weyl algebra's rigidity.

I am wondering if there is a nice presentation of the Hochschild cohomology of $A_n$ the Weyl algebra. It is known that $H^m(A_n,A_n)=0$ for $m>0$, and thus it is rigid. A proof can be found in ...

**1**

vote

**0**answers

24 views

### Homology of the subcomplexes of the “diamond shaped” sphere under 1-norm in $R^n$ as a simplicial complex

The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$.
The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial ...

**6**

votes

**0**answers

131 views

### kernel of the mod $2$ Bockstein on the first cohomology group

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...

**3**

votes

**0**answers

114 views

### Cocycle condition for 2-groups

I know that if $\omega_d(g_1, \ldots, g_d)$ is an d-cocycle characterized by $H^d(G,U(1))$, it satisfies the co-cycle condition
$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...

**9**

votes

**2**answers

206 views

### non-triviality of the underlying real vector bundle of the complexification of a real vector bundle

Let $M$ be a given manifold and $\xi$ be a given $k$-dimensional vector bundle over $M$. How to determine whether the underlying real vector bundle of $\xi\otimes\mathbb{C}$, i.e. the Whitney sum ...

**35**

votes

**10**answers

10k views

### 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 ...

**16**

votes

**1**answer

380 views

### A spectral sequence for computing cohomology of a space from that of its strata

Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D_1$, $D_2$, ..., $D_N$. For a set of indices $I$, let $D_I = \bigcap_{i \in ...

**5**

votes

**0**answers

91 views

### cohomology ring of configuration spaces on $S^2$ and the projective plane

For a manifold $M$ and a positive integer $n$, the unordered configuration space $B(M,n)$ is the space consisting of all unordered collections of $n$ distinct points on $M$. Precisely,
$$
...

**4**

votes

**0**answers

47 views

### Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher

It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a ...

**3**

votes

**0**answers

65 views

### Have locally principal crossed homomorphisms been studied?

Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...

**18**

votes

**1**answer

467 views

### Flat versus etale cohomology

Although the definition of etale ($\ell$-adic) cohomology is scary, I have at least some intuition for how it should behave: for instance, when it makes sense, I expect that it should be ``similar'' ...

**1**

vote

**1**answer

251 views

### Global sections of higher direct image sheaf

Let $X$ be a regular projective (complex) surface and $S$ be a finite set of closed points on $X$. Denote by $j:X\backslash S \to X$ the open immersion. Assume further that $H^1(\mathcal{O}_X)=0$. Is ...

**8**

votes

**2**answers

305 views

### Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...

**20**

votes

**1**answer

266 views

### Primary obstruction to the existence of a cross-section of $V_{n - q}(\omega)$ is a cohomology class in $H^{2q+2}(B, \pi_{2q+1} V_{n - q}(F))$?

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q + ...

**3**

votes

**1**answer

105 views

### What is known about the cohomology of the relative tangent bundle on a conic bundle?

Let us assume $X$ is a smooth, projective and unirational variety of dimension $n$ over $\mathbb{C}$.
Given a conic bundle $\pi: Y\rightarrow X$ such that $\omega_{\pi}^{-1}$ is relatively very ample ...

**43**

votes

**2**answers

1k views

### $H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
...

**4**

votes

**0**answers

79 views

### Example request: seriously deficient homogeneous spaces

In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...

**4**

votes

**0**answers

64 views

### A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...

**9**

votes

**0**answers

303 views

### Milnor-Stasheff Characteristic Classes Problem 7B, Borel 1953

There is the following Proposition 11.1 from Borel's 1953 paper La cohomologie mod 2 de certains espaces homogènes (see here).
Proposition 11.1 The classes $w^i$ and $\overline{w}^j$ are related ...

**2**

votes

**0**answers

85 views

### A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces.
The Künneth-Theorem which I ...

**6**

votes

**0**answers

143 views

### Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$:
$e^*\Omega^1_{E/X}$ and ...

**2**

votes

**0**answers

41 views

### Does anyone know if there is a generalization of symplectic Kodaira dimension beyond 4-manifolds?

I'm aware that in algebraic geometry, one has the Kodaira-Iitaka dimension, which generalizes the Kodaira dimension, but does anyone know if a correspondent generalization in the symplectic category ...

**3**

votes

**1**answer

133 views

### A converse to Whitehead's Second Lemma (and more)

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{h}$ be a finite dimensional $k$-Lie algebra. I'm interested in knowing which (Lie algebraic) properties $\mathfrak{h}$ ...

**3**

votes

**0**answers

157 views

### Cohomology algebra generated by $n$ Steifel whitney classes and and $k$ dual classes subject only to $n+k$ defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...

**3**

votes

**1**answer

255 views

### Hypersurfaces without variable cohomology

Let $X$ be a smooth projective variety over $\mathbb C$ of dimension $n+1$. If $Y$ is a smooth very ample hypersurface, we know that except $H^n(X;\mathbb Q)\rightarrow H^n(Y;\mathbb Q)$, the ...