A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-...

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2
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1answer
85 views

Normalization of Hochschild cocycles

Let $A$ be a unital algebra over $\mathbb{C}$. Let $C^n(A)$ be the space of all $n+1$-linear maps $f:A^{n+1} \to \mathbb{C}$ (to be called $n$-cochains). Define $b:C^n(A) \to C^{n+1}(A)$ by the ...
1
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0answers
65 views

Question on Hochschild cohomology

Let given ring $A$ without zero divisors and subring $\mathbf{k}\subset Z(A)$. Let also given $M~-$ $A~-$ bimodule, such that $xm = mx, \forall x\in \mathbf{k}, m\in M$. Is it true that if for $\...
6
votes
1answer
422 views

Continuous and smooth Lie groupoid cohomology

In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential $\...
1
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0answers
101 views

cohomology ring of homogenous manifold

Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces : $$ G/H ...
3
votes
1answer
77 views

What is known about the morphism $H^*_{Lie}(L,L)\to H^*_{Lie}(L,UL)$ induced by $L\hookrightarrow UL$

Let $L$ be a (differential) graded Lie algebra over a field $k$ of characteristic 0, and let $UL$ be the universal enveloping algebra of $L$. The inclusion $L\hookrightarrow UL$ induces a morphism of ...
5
votes
1answer
177 views

Bott-Samelson construction of a perfect Morse function on G/T

An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of ...
2
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0answers
72 views

When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$. By Example of a Schur-nontrivial group with no abelian subgroup ...
1
vote
1answer
133 views

Riemann-Roch for reducible surfaces

Let $C$ be a projective connected (reducible) curve over an algeraically closed field with nodes as singularities and $X=\mathbb P(\mathcal E)$ a projective bundle over $C$ (we know a ...
8
votes
1answer
145 views

Cohomology operations on unoriented cobordism

In unoriented cobordism there exist stable cohomology operations looking similar to Steenrod squares (they were used by Quillen to compute the unoriented cobordism ring with its formal group law ...
0
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0answers
132 views

Ring structure on cohomology of groups

Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter ...
4
votes
1answer
217 views

Hodge decomposition for Bott-Chern cohomology

$\DeclareMathOperator{im}{im}$ I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...
1
vote
0answers
49 views

computing homology of subvarieties of Euclidean spaces by persistent homology

Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$. Suppose the ...
1
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0answers
72 views

derived invariants, perversity and modular coefficients

Let $\pi:X\rightarrow Y$ a Galois cover of finite type schemes over $\mathbb{C}$ of group $\Gamma$. Let $n$ an integer such that it is not prime with the order of $\Gamma$. Then $\pi_{*}\mathbb{Z}/n\...
14
votes
2answers
508 views

List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds: Although it looks like a rather innocent technical statement, it is crucial for ...
0
votes
0answers
80 views

Action of a lattice on abelian varieties

Let $\pi\colon Y\to\mathbb{P}_\mathbb{C}^1$ be a ramified cover of degree two of $\mathbb{P}_{\mathbb{C}}^1$ such that $Y$ is smooth. I fix a point $x$ on $\mathbb{P}^1$, over which the cover is étale,...
4
votes
1answer
146 views

Computation on an Euler character

In the Bridgeland's paper "Flops and derived categories" (proof of (4.6), page 12), he computed an Euler character without much explanation. I thought this might not be difficult (and might not be ...
3
votes
1answer
254 views

Stiefel-Whitney classes of tensor product $\xi^m \otimes \eta^n$, computation

Let $\xi^m$ and $\eta^n$ be vector bundles over a paracompact base space. Where can I find a reference to the Stiefel-Whitney classes of the tensor product $\xi^m \otimes \eta^n$ being computed as ...
12
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1answer
500 views

Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...
0
votes
1answer
71 views

Is $H^{1}_{Sel}\left(K,E_{p^{n}}\right)\rightarrow\prod_{q \nmid \infty} H^1\left(K_{q},E_{p^{n}}\right)$ an injection?

If $K$ is a number field, $E$ an elliptic curve and $p$ a prime, does the Selmer group $$H^{1}_{\operatorname{Sel}}\left(K,E_{p^{n}}\right)$$ always inject into $$\prod_{q \text{ a nonarchimedean ...
6
votes
0answers
97 views

mod $p$ homology module of unordered configuration spaces of the projective plane

Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be ...
12
votes
1answer
295 views

Measuring the failure of pushforward to commute with Steenrod squares

Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general ...
57
votes
8answers
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
1answer
267 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 $H^*(M/G;F)$...
6
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0answers
112 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
1answer
321 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
10answers
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
3answers
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
0answers
120 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, \mathbb{...
6
votes
1answer
185 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
2answers
359 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
2answers
200 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 $H_{sing}^*(X,A)$;...
16
votes
2answers
356 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
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2answers
212 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
0answers
379 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
1answer
219 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
0answers
72 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
0answers
50 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
0answers
127 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
0answers
196 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
1answer
170 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
2answers
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 "...
11
votes
1answer
392 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
8answers
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
1answer
225 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 http://...
2
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0answers
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
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0answers
79 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
2answers
893 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 ...
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0answers
25 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
0answers
132 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
0answers
120 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}) + ...