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Questions tagged [cohomology]

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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Topology of connected subsets of the $3$-torus

Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$. We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded. I am ...
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80 views

Is the following variant of the Universal Coefficient Theorem valid?

A version of the Universal Coefficient Theorem that relates the integer cohomology of a group $G$ to its cohomology with coefficients in an abelian group $M$ is as follows: $H^n(G,M) = H^n(G,\mathbb ...
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110 views

About Hom and weight space of nilpotent Lie algebra cohomology

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote by $\Phi$ the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ ...
3
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1answer
79 views

Convergence of the Lyndon-Hochschild-Serre spectral sequence as an algebra

Consider a short (not necessarily split) exact sequence of groups $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ and suppose we wish to find the cohomology of $G$ with coefficients in a ...
11
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1answer
347 views

Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology?

There are numerous expositions of simplicial homology in the literature. Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes. Hatcher in “Algebraic ...
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1answer
213 views

What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian?

Let $\operatorname{Gr}(k, n)$ and $\operatorname{Gr}^+(k, n)$ denote the unoriented and oriented grassmannians respectively. The $\mathbb{Z}_2$ cohomology of the unoriented grassmannian is $$H^*(\...
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83 views

Cohomology of toric blowup

Let $n\geq2$. Let $G$ be a linear automorphisms group of prime order on $\mathbb{C}^n$. We assume that $0$ is the unique fixed point of $G$. I consider the quotient $\mathbb{C}^n/G$. It is a toric ...
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0answers
62 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
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1answer
147 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 $k$-...
19
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1answer
194 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, \...
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0answers
160 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 ...
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2answers
1k 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
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1answer
90 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
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0answers
91 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 $...
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438 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 ...
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0answers
192 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 ...
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0answers
48 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
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1answer
232 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 ...
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73 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
1answer
161 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 ...
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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 ...
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109 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 ...
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0answers
122 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 ...
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0answers
57 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})...
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“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 ...
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0answers
69 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 $...
11
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1answer
214 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
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1answer
407 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
3answers
322 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
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1answer
162 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
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0answers
43 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 $X$ ...
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1answer
132 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
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1answer
708 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
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1answer
171 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
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1answer
107 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
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1answer
247 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 ...
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0answers
48 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
1answer
439 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 ...
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0answers
70 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 ...
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1answer
252 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 $...
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1answer
114 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 ...
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0answers
124 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 ...
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183 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 $P\rightarrow M$ ...
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0answers
177 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 ...
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1answer
486 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 ...
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0answers
103 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 ...
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1answer
99 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 (...
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1answer
443 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
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1answer
590 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
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0answers
87 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-...