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-...

**1**

vote

**0**answers

17 views

### Order relation between cohomology groups

We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex
$$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...

**3**

votes

**1**answer

130 views

### Known techniques to compute flat cohomology after base change

Let $f$ be some homogenous polynomial of degree $d>2$.
Let $X = \operatorname{Proj} (k[x,y,z]/(f))$ where $k$ is algebraically closed field of characteristic $p >0$.
Now let $R$ be a $k$-...

**15**

votes

**0**answers

91 views

### To what extent can we characterise the image of the topological Chern character?

For a finite CW complex $X$, the Chern character gives an isomorphism
of finite-dimensional vector spaces:
$$
ch : K^*(X)\otimes \mathbb{Q} \to H^*(X, \mathbb{Q}).
$$
The vector space $V = H^*(X, \...

**-1**

votes

**0**answers

66 views

### A generalization of Conner Conjecture

Let $G$ be a compact (abelian) totally disconnected group and $X$ be a
compact $G$-space. If $X$ is $%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
$-acyclic space (i.e. $\widetilde{...

**2**

votes

**0**answers

153 views

### Torsion free-ness of cohomology of moduli of vector bundles

My question requires a little introduction:
$\textbf{Atiyah-Bott's solution:}$ In the paper "The Yang mills equation of Riemann surfaces" Atiyah-Bott has computed the cohomology of moduli vector ...

**20**

votes

**2**answers

947 views

### Hodge theory (after Deligne)

In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely ...

**4**

votes

**1**answer

78 views

### Correspondence Between First Galois Cohomology and Semilinear Actions Up to Isomorphism

I've been stuck for a while on Exercise 1.9 of Bjorn Poonen's "Rational Points on Varieties". We start with $L/K$ a finite Galois extension with Galois group $G$, some $r \in \mathbb{Z}_{\geq 0}$, and ...

**3**

votes

**0**answers

83 views

### Cohomology groups for singular non-compact variety with paving by affine spaces

Suppose $X$ is a complex algebraic variety that is paved by affines. We take the most general definition, which is that $X$ has a filtration $0=X_0 \subset X_1 \subset \cdots \subset X_n=X$, each $...

**17**

votes

**0**answers

419 views

### Which rings are cohomology rings?

Which rings can arise as cohomology rings of algebraic varieties?
To be more specific, take a Weil cohomology theory $H^*$ with coefficients in a field $K$ of characteristic 0 defined for smooth ...

**9**

votes

**0**answers

167 views

### Continuous cohomology of a profinite group is not a delta functor

Let $G$ be a profinite group, then there is a general notion of continuous cohomology groups $H^n_{\text{cont}}(G, M)$ for any topological $G$-module $M$ (I require topological $G$-modules to be ...

**3**

votes

**0**answers

47 views

### Comparing the cohomology rings of two central extensions

Consider two groups $G$ and $G'$, where $G$ is the direct product of groups $A$ and $B$, with $B$ abelian, and $G'$ is a nontrivial central extension of $A$ by $B$. Suppose that as groups, $H^1(G,M) \...

**6**

votes

**1**answer

228 views

### “Cyclic” continuum

On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition:
"A curve is said to be cyclic if its first Čech cohomology group with integer coefficients ...

**0**

votes

**0**answers

72 views

### Local cohomology with disjoint support

Let $X$ be a topological space, $Z_1, Z_2$ two disjoint subspaces of $X$. Let $F$ be a sheaf of abelian groups on $X$. Is it true that for any $i \ge 0$,
$$\mathrm{Im}(H^i_{Z_1}(X,F) \to H^i(X,F)) \...

**4**

votes

**1**answer

153 views

### Hodge decomposition of the symmetric product of a curve

Let X be a smooth projective connected curve over $\mathbb{C}$ and let $n>1$ be an integer. Let $Y= Sym^n_X$ be the $n$-th symmetric product of $X$.
Is there, for every $i$, a nice formula ...

**1**

vote

**0**answers

25 views

### A regular sequence in a quotient by a “half lattice” defined by a toric manifold

I am interested in some properties of polynomial algebras associated with smooth compact toric varieties. Recall that a toric manifold can be obtained as a quotient $$P^{-1}(p) / \mathbb{K}$$ by the ...

**2**

votes

**0**answers

108 views

### On finite dimensional commutative algebras and regular sequences

Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...

**5**

votes

**0**answers

115 views

### A Poincare dual spectral sequence for equivariant cohomology: extension to generalized cohomology?

This question is a follow-up to my previous question:
"Rotated" version of the Atiyah-Hirzebruch spectral sequence
In that question, I discussed two different spectral sequences for ...

**2**

votes

**0**answers

56 views

### On cohomological algebras related to toric manifolds

I am interested in some cohomological algebras related to toric manifolds. We consider a toric manifold $M$ as a quotient
$$M = P^{-1}(p) / \mathbb{K}, \quad P : \mathbb{C}^n \to \text{Lie}(\mathbb{K})...

**8**

votes

**0**answers

232 views

### “Complementarity” between homotopy and cohomology [duplicate]

I was browsing MO and I have stumbled upon this answer which discusses why we should expect homotopy groups of spheres to be complicated. One heuristic argument given is that "the theory needs to have ...

**0**

votes

**0**answers

66 views

### On the dimension of the cohomology of toric manifolds

Let $M$ be a toric manifold. I'm not sure what conditions on $M$ are required, but one can assume, if needed, that it is compact, smooth, etc. We consider $M$ as a quotient given by the momentum map $...

**10**

votes

**1**answer

168 views

### Is there a kind of Poincare duality for Borel equivariant cohomology?

Let $G$ be a finite (or discrete) group, $M$ a $d$-dimensional manifold with smooth $G$-action (I am interested in the case where the action is not free, so $M/G$ is not a manifold). For an Abelian ...

**13**

votes

**1**answer

390 views

### Is there an explicit Dold-Thom theorem?

The Dold-Thom theorem tells us that we can recover the reduced homology of a pointed space $(X,x)$ via taking homotopy groups of the symmetric product:
$$\pi_i(\mathrm{Sym}^{\infty}(X,x)) \cong H_i(X,...

**4**

votes

**3**answers

299 views

### Outline of the proof that Cech cohomology and singular cohomology coincide on any locally contractible space

If $X$ is paracompact and locally contractible, then singular cohomology and Cech cohomology of $X$ coincide, with coefficients in any abelian group.
I hear that this is a classical result but I fail ...

**3**

votes

**1**answer

152 views

### Poincaré dual of the generators of $H^d(\mathbb{RP}^5,\mathbb{Z}_2)$

We know $H^d(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2$. So there are two classes of $\mathbb{Z}_2$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$.
Wha are the Poincaré dual $(5-d)$-...

**3**

votes

**0**answers

42 views

### Irreducible separators of compact manifolds

Definition. A closed subset $S$ of a topological space $X$ is called
$\bullet$ a separator of $X$ if $X\setminus S$ is disconnected;
$\bullet$ an irreducible separator if $S$ is a separator of $X$ ...

**1**

vote

**1**answer

126 views

### Commutativity between functors on sheaves of abelian groups

I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...

**18**

votes

**1**answer

687 views

### Double Counting: Motivic Edition

One of the most important proof techniques in combinatorics is double counting: proving that both sides of an identity count elements of some set in two different ways. This question is an attempt at ...

**2**

votes

**1**answer

169 views

### Coefficient (or target) category for factorization homology

In the article "Factorization homology of topological manifolds" by Ayala and Francis, a symmetric monoidal $\infty$-category $\mathcal{V}$ is fixed as the target or coefficient category. This ...

**2**

votes

**1**answer

104 views

### Construction of differentials in the spectral sequence for double complexes

I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d_2$ differentials of the spectral sequence (...

**7**

votes

**1**answer

222 views

### Cohomology of $\mathbb Z_4$ via the Lyndon-Hochschild-Serre spectral sequence

I'm trying to understand how to construct the Lyndon-Hochschild-Serre spectral sequence for the cohomology (with integer coefficients) of the central extension $G$ of a group $Q$ by a group $N$, given ...

**1**

vote

**0**answers

47 views

### Direct limit of complexes from cocycle property up to homotopy

Suppose that we have a directed set $(I, \leq)$, and a set of maps
\begin{equation}
f_{i,j} : C^*(A_i) \to C^*(A_j), \quad i \leq j,
\end{equation}
of singular cochain complexes of topological spaces $...

**8**

votes

**1**answer

425 views

### What is the interpretation of the Gerstenhaber bracket?

The homology of an $E_2$-algebra is a Gerstenhaber algebra.
How precisely is the Gerstenhaber structure related to the $E_2$-structure?
Obviously, the Gerstenhaber product is the commutative product ...

**3**

votes

**0**answers

63 views

### Convolution of sheaves on R

I am trying to understand a basic computation of convolution. Throughout, $R$ is the real line as a topological group and $k$ is some base field. I would like to understand the computation of the ...

**-5**

votes

**1**answer

247 views

### Computing $H^*(BDiff(W_{\infty},D^{\infty});\mathbb{Q})$ via Mumford-Morita-Miller classes

Galatius and Randal-Williams proved the following generalized Mumford conjecture in their joint paper, "Stable Moduli spaces of High Dimensional Manifolds". For each characteristic class of oriented $...

**1**

vote

**1**answer

106 views

### Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?

In Witten's paper Three Dimensional Gravity Revisited and Quantization of Chern-Simons Theory with Complex Gauge Group, he used a fact that for a principal $G$-bundle, the quantization of the Chern ...

**3**

votes

**0**answers

118 views

### Cohomology of the Hilbert square of a degree 14 K3 surface [Beauville-Donagi]

I have a question about the article by Beauville-Donagi called La variété des droites d'une hypersurface cubique de dimension 4 (C. R. Acad. Sc. Paris, t. 301, Série I, n° 14, 1985).
Their ...

**8**

votes

**0**answers

167 views

### Twisted Chern-Simons, and Twisted Wess-Zumino Term

I am asking this question about Chern-Simons theory from the paper "Quantum Field Theory and Jones Polynomial" by Edward Witten.
Let $M$ be a closed three dimensional manifold, and $P\rightarrow M$ ...

**9**

votes

**0**answers

164 views

### cohomology of flag variety

I recently ran into a 30+ years old literature by Andersen and Jantzen on some calculations on cohomology of flag varieties (Cohomology of Induced Representations for Algebraic Groups). Here is the ...

**0**

votes

**1**answer

484 views

### How do we compute the even cohomology $H^{2i}(Q)$ of the affine hyperquadric?

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.
What is a reasonable Kähler metric for $Q$ (induced by the ...

**3**

votes

**0**answers

98 views

### $\partial \overline{\partial}$-lemma for Irreducible, Normal Projective Varieties

Reference: W. Ding, G. Tian -- Kähler--Einstein metrics and the Generalised Futaki Invariant, Inventiones mathematicae, (1992).
Let $X$ be a normal projective variety which is irreducible. Given an ...

**0**

votes

**1**answer

96 views

### Compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of a product of Lie groups (and their quotients)

As the following product is a bit unfamiliar to me:
How do we compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of the product of Lie groups:
$M=SO(n_1)\times U(n_2)\times SU(n_3)\times (...

**11**

votes

**1**answer

430 views

### Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness

Suppose that we are given a topological space $X$: assume for simplicity that $X$ is compact we want to adress the following question:
Is it true that one can find a manifold $M$ which is homotopy ...

**2**

votes

**1**answer

587 views

### Show that the rational cohomology ring $H^*(M;\mathbb{Q})$ needs at least two generators

Let $M$ be a simply connected closed Riemannian manifold. How does one find a necessary condition going both ways that may be imposed on $M$ (perhaps on the curvature of $M$ and on torsion) which ...

**5**

votes

**0**answers

83 views

### Restricting projective representations of Lie groups to lattices

Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...

**4**

votes

**1**answer

118 views

### Trivial cohomology with free module coefficient

Let $G$ be a group and $M$ be a free $\mathbb{Z} G$-module. Then $H^2(G,M)=0$. Is this statement correct?
I know that if $M$ is injective module, then $H^n(G,M)=0$ for all $n\geq 1$. But I have no ...

**3**

votes

**0**answers

49 views

### cohomology of the orbit space of a compact totally disconnected group action on a paracompact space

It is well-known the next theorem at Chapter III, Theorem 7.2. in Bredon's Introduction to compact transformation groups book.
Theorem: Let $X$ be a paracompact $G$-space with $G$ finite and let $\pi:...

**4**

votes

**0**answers

235 views

### A cohomology associated to a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold.
Let $$\Omega_{\omega}^k(M)=\{\alpha \in \Omega^{k}(M)\mid \alpha \wedge \omega \;\;\text{is an exact form}\}$$
Then we have a chain comlex$$\...

**8**

votes

**1**answer

348 views

### What is closed homology?

Bott & Tu in Differential forms in Algebraic Topology write in Remark 5.17, pg.52
The two Poincare duals of a compact orientated submanifold correspond to two homology theories - closed and ...

**4**

votes

**0**answers

300 views

### Relative cohomology in algebraic topology vs algebraic geometry

There is a notion of relative cohomology in both algebraic topology and algebraic geometry and the flexibility granted by this relative viewpoint is quite powerful in both cases. However, the two ...

**15**

votes

**1**answer

422 views

### “Rotated” version of the Atiyah-Hirzebruch spectral sequence

Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-...