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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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55 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
41 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
301 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
59 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
202 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
101 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
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0answers
109 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
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0answers
155 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
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0answers
155 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 ...
1
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1answer
466 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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93 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
89 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
417 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
562 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
75 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
1answer
112 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 ...
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46 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
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0answers
229 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
1answer
339 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 ...
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281 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 ...
14
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1answer
378 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 $G$-...
3
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1answer
205 views

Cohomology with compact support

Let $X$ be a smooth, projective variety, $V_1, V_2$ smooth, closed subvarieties of the same dimension and $E$ a locally free sheaf on $X$. There exist natural morphisms $$r_1: H^i_{V_1}(E) \to H^i(E) ...
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1answer
178 views

Kähler form on complex projective algebraic variety [closed]

I am not very familiar with the notion of projective algebraic varieties, I work mostly from an algebraic topology/differential geometry point of view, but I am trying to find a prove for the ...
2
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1answer
145 views

Morava $E$-theory source of reading

Recently I came across a paper which I really need to read through and which uses language of Morava $E$-theory. Since I'm not comfortable with this cohomology theory, I've been looking for quite a ...
3
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2answers
331 views

Milnor's proof of cohomology of BO(n)

In Milnor/Stasheff characteristic classes there is the proof that $H^*(BO(n);\mathbb{Z}_2)$ is the polynomial ring on the first n Stiefel-Whitney classes. I understand the part that the latter ring is ...
2
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0answers
97 views

How to show the equality of two descriptions for the cohomology of a non-finite group

I am learning about group cohomology. For a group $G$ and a $G$-mod $A$, we can define $X^n(G,A)=Map(G^{n+1},A)$, and get a resolution $0\to A\to X^\cdot$ and then define cohomology groups $H^n(G,A)$...
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40 views

Fibre restriction and Leray–Hirsch

Let $\mathbb{C}P^h\to E\to B$ be a bundle which is cohomologically trivial via Leray–Hirsch, i.e. if we denote $H^*(\mathbb{C}P^h)\cong \mathbb{Z}[\mu]/\mu^{h+1}$, we have a class $\chi\in H^2(E)$ and ...
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0answers
70 views

Extending Beauville-Bogomolov orthogonal decomposition from variety to scheme

I'm seeking to understand the de Rham cohomology of a Hilbert scheme $K3^{[4]}$ of the K3 surface. By Beauville, this 8-dimensional compact manifold is Kaehler, irreducible, holomorphically symplectic ...
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205 views

(pro)Étale cohomology of adic spaces and inverse limit

I am studying Peter Scholze's paper $p$-adic Hodge theory for rigid-analytic varieties and I am confused by the following. Let $X$ be a finite type scheme over $\mathbb{C}_p$ (proper and smooth if ...
4
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1answer
134 views

Hochschild homology of a category of modules over an algebra

Suppose $A$ is an algebra over some field, say the complex numbers if that helps. Then we can consider the category $\mathbf{C}_A$ of finite-dimensional modules over $A$. This category can be seen as ...
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80 views

Integral cohomology on compact irreducible holomorphic symplectic manifolds

In particular, I'm attempting to understand the integral cohomology on K3^[4], a Hilbert scheme on the projective K3 surface, and whether or not this compact Kaehler manifold admits a Kaehler form ...
2
votes
2answers
207 views

Projective surfaces with vanishing first cohomology

Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $D$ an effective divisor in $X$. Is it true that $H^1(\mathcal{O}_{X\backslash D})=0$ (we know that $H^1(\mathcal{O}_X)=0$)? If not true ...
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123 views

Explicit elements of the first cohomology of modular curves

Let $M$ be a modular curve and $\pi:E\to M$ the universal elliptic curve. For a prime $\ell$, let $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$. I am wondering whether there are any explicit ...
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1answer
125 views

Torsion in the integral cohomology of $BPU_{n}$

I would like to prove that the integral cohomology of $BPU_{n}$ the classifying space of the projective unitary group of order $n$ has $n-$primary torsion. We have a fiber sequence of the form $BSU_{...
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1answer
226 views

Cup products in the Mayer-Vietoris sequence

Let $(X;U,V)$ be an excisive triad and consider the corresponding part of the Mayer-Vietoris sequence $H^{\bullet-1}(U\cap V)\stackrel{\delta^*}{\to} H^\bullet(X)\to H^\bullet(U)\oplus H^\bullet(V)$. ...
6
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1answer
388 views

Why Use Hypercohomology When Defining the de Rham Cohomology of a Smooth Scheme over $k$?

Hopefully this question is of an appropriate level for this site: I'm reading some notes by Claire Voisin titled Géométrie Algébrique et Géométrie Complexe. Let $X$ be a smooth $k-$scheme. In these ...
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101 views

finiteness of syntomic cohomology of proper varieties over algebraically closed and finite fields

Let $k$ be an algebraically closed field or a finite field and $X/k$ be a proper variety. Let $G/X$ be a finite flat group scheme. Is $\mathrm{H}^i_\mathrm{syn}(X,G)$ finite for all $i$? (So: are ...
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182 views

étale vs syntomic vs flat cohomology

Let $\mathscr{A}/X$ be an abelian scheme over $X$ of characterisitic $p$. The étale topology is not fine enough for the Kummer sequence for $\mathscr{A}$ to be (right) exact, but the syntomic and flat ...
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0answers
208 views

A cohomology associated to a Riemannian manifold

Let $N$ be a compact Riemanian manifold and $G$ be its isometry group. Let $M=\chi^{\infty}(N)$ be the space of smooth vector fields on $N$. There is a natural right action of $G$ on $M$ with $X.g=g^*(...
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votes
2answers
203 views

Minakshisundaram-Pleijel zeta function identity

Let $\zeta(M,s)$ be the Minakshisundaram-Pleijel zeta function, which encodes the eigenvalues of the Laplace-Beltrami operator. Where can I find a proof or reference of the following identity? If $M$ ...
6
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1answer
192 views

Which compact (finite dimensional) Lie groups have $H^1_{DR}(G)\neq 0$

In particular, I am wondering if $H^1_{DR}(G)\neq 0$ implies that the group can written as a semidirect product of $\mathbb{S^1}$ and something else, with the $\mathbb{S^1}$ factor being responsible ...
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0answers
70 views

Pull back group cohomology onto handle decomposition

A construction encountered in the Dijkgraaf-Witten invariant uses the following ingredients: An oriented, (assumed here to be smooth) manifold $M^n$ A finite group $G$ (and a field, chosen to be $\...
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42 views

What is known about the cohomology of the matrix monoid?

When I say the cohomology of a monoid, I mean that of its classifying space (considering the monoid as a category with a single object). Let $M_n(R)$ be the monoid of matrices with matrix ...
2
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1answer
60 views

Description of Koszul dual of Sklyanin algebras

It is well-known that Sklyanin algebras are Koszul, but, is it known an explicit description of the dual algebra Ext_A(k,k)? (I mean in terms of generators and relations)
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123 views

Cap product for (co)homology from handle decompositions/Kirby diagrams

Since handle decompositions and Morse functions are intimately related, I'm imagining that a given explicit handle decomposition allows for an explicit description of the cellular complex and thus of (...
12
votes
1answer
554 views

Cohomology of $ko,tmf,MSpin,MString$ with coefficients $\mathbb{Z}/p$ for odd primes $p$

It is well-known that $$H^*(ko,\mathbb{Z}/2)=\mathcal{A}\otimes_{\mathcal{A}(1)}\mathbb{Z}/2$$ $$H^*(tmf,\mathbb{Z}/2)=\mathcal{A}\otimes_{\mathcal{A}(2)}\mathbb{Z}/2$$ where $\mathcal{A}$ is the mod ...
2
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1answer
150 views

Linear sections of $Gr(V,2)$

Let $V$ be a vector space, and consider $G=Gr(V,2)\subset \mathbb{P}^N$ embedded via the Plucker embedding. Let $W\subset \mathbb{P}^N$ be a linear subspace. I want to find the class $[W\cap G]\in A(G)...
2
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1answer
325 views

What is the scope of validity of Kunneth formula for de Rham?

In books like Bott-Tu or all pdf texts I have found on internet, the Kunneth formula for manifolds $M$ and $N$ and their de Rham cohomology $$ H^{\bullet}_{dR}(M \times N) \simeq H^{\bullet}_{dR}(M) \...
3
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0answers
116 views

Finding generators of equivariant cohomology

Let $(M,\omega)$ be a symplectic manifold with symplectic form $\omega$, carrying a Hamiltonian action of a compact connected Lie group $G$ with moment map $\mu:M\to \mathfrak{g}^\ast$, where $\...
2
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59 views

Do we have a transgression formula for the chern characters of quasi-isomorphic cochain complexes of vector bundles?

Let $(E^{\cdot},d_E^{\cdot})$ be a cochain complex of complex vector bundles on a smooth compact manifold $X$. Now for each $E^i$ we could assign a connection $\nabla_E^i$ and obtain its curvature $(\...