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-cohomology, motivic-cohomology, equivariant-cohomology, ...
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Compactly supported cohomology of a topological DM stack
Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally ...
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Structure of the variety of $n$-tuples of $m \times m$ matrices with zero product
Consider the functor sending a commutative ring $R$ to $\{(A_1,\dots,A_n) \in ( M_m(R) )^n | A_1 \dots A_n =0 \}$ which defines a scheme over $\mathbb Z$, let $X$ be its base change to $\mathbb C$.
...
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Pushforward in Compactly Supported Cohomology
Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
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3rd Cohomology of a fibration with Flag varieties as fibers
Let $X$ be a smooth projective rational variety over $\mathbb{C}$, let $Y$ be another smooth projective variety, both of dimension bigger than 2, and let $\pi : Y \rightarrow X$ be a locally trivial ...
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cohomology of curves
Let $X$ be a smooth projective complex curve. Consider the diagonal $\Delta$
in $X \times X$, and $\mathcal{O}(\Delta)$ the associated line bundle.
If $j$ is the inclusion of $\Delta$ in $X \times X$ ...
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$d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$
Cross posted from here after no responses and a bounty being placed on the question.
Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-...
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Computation of mod p homology of $MSU$
I am trying to proof Novikov theorem
\begin{equation}
MSU_*\otimes \mathbb Z[\frac 1 2] \cong \mathbb Z[\frac 1 2][y_2, y_4, \ldots],\quad \deg y_i = 2i.
\end{equation}
This can be proved by using ...
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Unifying "cohomology groups classify extensions" theorems
It is common for the first or second degree of various cohomologies to classify extensions of various sorts. Here are some examples of what I mean:
1) Derived functor of hom, $\text{Ext}^1_R(M, N)$. ...
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A question about Johnson's theorem on the first and second cohomology of commutative amenable algebras
Johnson in cohomology of Banach algebra proved the following proposition.
I need to some guidance for the bold part of the following proof. Do you know any papers or book with more details for this ...
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Easier ways to compute homology/cohomology by adding extra structure
Suppose $X$ is a topological space and I want to talk about its “homology”.
There is this notion of singular homology obtained from singular chain complex. This is not very easy to compute.
Suppose ...
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Kaplansky Idempotent conjecture and Extension theory
We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...
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Stable homology operations
Let $x\in (H\mathbb F_2)_n(X)=[S^n,H\mathbb F_2 \wedge X]$ be a homology class for a space $X$. Is there a description of
$$[S^n\overset x\to H\mathbb F_2 \wedge X\overset{Sq^r\wedge id}\to \Sigma^r H\...
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"a sign that one should be computing K-theory"
Allen Knutson said here in comments below the question that
I generally regard torsion in (co)homology as a sign that one should be computing K-theory instead, which has less of it.
I know one ...
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Image of a map on cohomology rings
The following seems like an extremely basic algebraic topology question, but it's not something I ever learned, nor does it look familiar to the algebraic topologists I've asked.
Let $f:X\to Y$ be ...
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On normalized 2-cocycle
Let $G$ be a group acts trivially on an abelian group $A$. Let
$\varepsilon $ be a normalized 2-cocycle in $ Z^{2}(G,A)$. Assume
that $G=H_{1} \times H_{2}$ and let $\varepsilon_{1}=res_{H_{1}\times
...
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Simply put Floer homology
I would like to understand what exactly Floer homology $HF_*(Y)$ for a simply connected compact 3-manifold $Y$ is. I understand there are many variants of Floer homology (and cohomology) but I would ...
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Kähler manifold with even-only singular cohomology
Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all ...
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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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Lie algebra cohomology of loop algebra
Let $G$ be a simple algebraic group over the complex numbers. Is it true that the Lie algebra cohomology $H^*(L\mathfrak{g}, \mathbb{C})$ of the loop Lie algebra $L\mathfrak{g}=\mathfrak{g} \otimes \...
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Analogue of integration for group cohomology
Consider some oriented surface $S$ with fundamental group $\pi_1(S)$. The group cohomology of $\pi_1(S)$ with coefficients in $\mathbb{R}$ is isomorphic to the de Rham cohomology of $S$. In degree 2, ...
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Cup product in Tate Cohomology Ring
Let $G$ be a finite cyclic group of order $p$. The tate cohomology groups $\hat{H}^*(G, \mathbb{F}_p)$ are defined using a complete resolution of $\mathbb{F}$ as $\mathbb{F}_pG$-module.
There is a ...
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Generalized de Rham cohomology on product bundle giving specified cohomology
Given a compact, smooth manifold $M$ and a real vector bundle $E \to M$ (in general not flat). There already have been numerous questions about how to equip the space $\bigoplus_k \Gamma(\Lambda^k T^* ...
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Definition of the Gauss symbol [closed]
Tahara refers to the "Gauss symbol" in the article, On the second cohomology groups of semidirect products, Math. Z. 129 (1972) 365--379. For a fixed $n$, let $S_{ij}$ be the expression
\begin{...
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When simple cohomological computations predict ingenious algebro-geometric constructions?
Classical algebraic geometry is full of ingenious constructions and miraculous coincidences: 27 lines on a cubic surface are related to Weyl lattice of type $E_6,$ lines on an intersection of four-...
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A manifold is a homotopy type and _what_ extra structure?
Motivation: Surfaces
Closed oriented 2-manifolds (surfaces) are "classified by their homotopy type". By this we mean that two closed oriented surfaces are diffeomorphic iff they're homotopy ...
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Galois cohomologies of an elliptic curve
I asked this question on Mathematics Stack Exchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois ...
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Is $B\mathbb{G}_m$ strongly $A^1$-invariant?
I have just seen the definition of strongly ${A}_1$ invariance:
A sheaf of group $G$ is strongly $A_1$ invariance , if both $H^0(-;G)$ and $H^1(-;G)$ is $A_1$-invariant.
I haven't got too much ...
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Cohomology of sheaves on $X \cup_{Z} Y$
I am in the following situation, I have two schemes $X$, $Y$ and two closed immersions $Z \rightarrow Y$, $Z \rightarrow X$. Everything is smooth. I am interested in calculating morphisms in the ...
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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}).
$$
The vector space $V = H^*(X, \...
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What can be said about the topological K-theory of non-singular varieties of small codimension in projective space?
Working over $\mathbb{C}$, the Barth-Larsen results tell us a lot about the ordinary cohomology of non-singular varieties of small codimension in projective space. For example if $X \subseteq \mathbb{...
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Which $K$-groups $K(C^*_r(G))$ are computed?
We have the Pimsner-Voiculescu exact sequences and the Baum-Connes map
for possible computation of the $K$-theory of the reduced group $C^*$-algebra $C^*_r(G)$ for a topological, locally compact, ...
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Free DGA given a map and cohomology groups
Why do Free DGAs on a morphism often give the same (co)homology as other (co)homologies?
Here is the example that comes to mind first:
Example: Let $R$ be a ring, let $A$ be an $R$-algebra, let $M$ ...
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$L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles
First of all I want to apologize for the much too long post.
A Lie group $G$ is acting on a smooth manifold $M$, then we define
\begin{align*}
T^k_G(M)=
(S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...
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What is the smallest number $d$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d))$ vanishes?
Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line.
What is the ...
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Differential forms of a Lie group giving cohomology of the Lie group
Consider a manifold $M$. Then, we have the notion of differential forms on $M$ and complex associated to that, denoted by $$\cdots\rightarrow \Omega^{k-1}(M)\rightarrow \Omega^k(M)\rightarrow \Omega^{...
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cohomology of nilpotent matrices of fixed $m$-th power
Let $k$ be an algebraically closed field, $\mathcal{N}$ is the variety of $n \times n$ nilpotent matrices over $k$, and consider the natural $m$-power map $\mathcal{N} \rightarrow \mathcal{N}$ given ...
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De Rham cohomology of Lie groupoid
Let $G$ be a Lie group acting on a manifold $M$.
Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$. We have the notion of de Rham cohomology of a Lie groupoid by ...
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Second Bounded Cohomology of a Group: Interpretations
Suppose we have a group $\Gamma$ acting on an abelian group $V$. Then it is well-known that the second cohomology group $H^2(\Gamma,V)$ corresponds to equivalence classes of central extensions of $\...
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Coherent Sheaf supported in a point
Hi guys, I'm studying Cech cohomology of sheaves and I've the following doubt:
If you have a coherent sheaf $\mathcal{F}$ in a compact complex variety $X$ then the cohomology groups are finite ...
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$\partial \overline{\partial}$-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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Deformations of Vertex Algebras
As the title suggests, I'm interested in deformation theory of vertex algebras and their representations.
In the paper https://arxiv.org/abs/1806.08754, the authors construct, for a vertex algebra $...
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Homology of the universal cover
$k$ is a field. Let $X$ be a connected pointed $CW$-complex such that the homology
$H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous ...
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An exact complex of tensor families
Given a field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, some natural numbers $q,n,m$ with $q \leq n$ and $m < n$ and a polynomial algebra $S := \mathbb{K}[\eta_1, \dots, \eta_n]$. We can consider $...
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Sketch of Weil's proof of the Riemann hypothesis for curves
I was wondering if anybody could provide a sketch of Weil's proof of the Riemann hypothesis for curves that uses the Jacobian $\text{Pic}^0(X)$ and a bit of the intersection theory on $X \times X$ and ...
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A cohomology associated to a vector field on a Riemannian manifold
Edit: Accoring to the comment of Asura Path I revise the question.
Let $X$ be a vector field on a Riemannian manifold $(M,g)$. So we have a $1$-form $\beta$ with $\beta(Y)=\langle Y,X\...
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Comparing cohomology using homotopy fibre
I have a question, which might be very basic, but I don't know enough topology to answer.
Suppose you have a map of topological spaces (or homotopy types) $f : X \to Y$, with homotopy fibre given by ...
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Cokernel of section of a general coherent sheaf
Given a scheme $X$ and an $\mathcal{O}_{X}$-module $\mathscr{E}$, we know that a section $s \in H^{0}(X, \mathscr{E})$ is equivalent to a morphism $s :\mathcal{O}_{X} \to \mathscr{E}$. It is the ...
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Cohomology of $\operatorname{SO}(p,q;\mathbb{Z})$ with $p=3,q=19$
I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $K3$-surface. It is known that this space is given by the bi-quotient $SO(3,19;\mathbb{Z})\setminus SO(3,...
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Teaching cohomology via everyday examples
This question is a "sequel" to my similar questions about the fundamental group and homology. All of these questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics ...
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Hodge-theoretic criterion for smoothness
Let $k$ be an algebraically closed field of any characteristic. Is it possible to give an equivalent condition for a $k$-variety to be smooth using only the cohomology of the variety (and whatever ...
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