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3
votes
1answer
145 views

When is the cohomology of a space Ext(k,k)?

If $X$ is aspherical, we know that $H^*(X,k) = \text{Ext}_R(k,k)$, with $R = k\pi_1$. For non-aspherical spaces, do we ever have $H^*(X,k) = \text{Ext}_R(k,k)$ for some ring $R$? Obviously we need ...
3
votes
0answers
91 views

Action of automorphisms on cohomology with supports

Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich ...
3
votes
1answer
98 views

LES for relative cohomology via sheaves

I was unable to find a suitable answer for the following question: Once one learns that singular cohomology is the same as cohomology with coefficients in locally constant sheaf, it is natural to try ...
17
votes
1answer
419 views

What are explicit obstructions to realizability of formal group laws as complex-oriented ring spectra?

Recall that a complex-oriented spectrum is a ring spectrum E with a map $MU \to E$. Analogously, a ring with a (1-d commutative) formal group law is (represented by) a ring $R$ with a map $L \to R$ ...
9
votes
1answer
256 views
+100

Identifying a Hopf algebra cohomology theory

Here is a cohomology theory for a Hopf algebra, which I am sure has appeared elsewhere. I met it in the van Est spectral sequence for Hopf algebras. Apologies for my being stupid here, but it would be ...
6
votes
2answers
301 views

Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$

Let $X$ be a nice topological space and denote by $\pi_1(X)$ its fundamental group. It is well-known that there is a well-defined map $$ 0 \rightarrow H^2(\pi_1(X),A) \rightarrow H^2(X,A),$$ where ...
7
votes
1answer
228 views

What is an example of a formal group law that is Landweber-exact but not flat?

Quick Background: The $p$-series of $F$ (where $F$ is a formal group law over a graded ring $R$) will be of the form $[p](x) = px + v_1x^{p^1} + ... + v_nx^{p^n} + ...$ ; $(F, R)$ is Landweber-exact ...
0
votes
0answers
57 views

Cohomology operators inducing local basis of $1-$forms

Suppose that $\partial$ is a non-trivial ($\partial \neq 0$) cohomology operator on an $m-$dimensional manifold $M$ (that is: $\partial:\Omega(M)\to\Omega(M)$ is a degree $1$ derivation such that ...
1
vote
1answer
226 views

Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$. We have the ...
2
votes
1answer
156 views

When is finding an explicit inverse of an isomorphism not possible

My question is about Shapiro's lemma. Consider the isomorphism $\phi: H^n(G, Hom_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains. So the obvious map is ...
2
votes
0answers
80 views

Deformations of associative algebras and Hochschild cohomology

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer: Let $(A,\mu)$ be a commutative associative algebra ...
2
votes
1answer
151 views

Galois cohomology out of the classifying stack

Suppose $G$ is a smooth and abelian $k$-group scheme, for $k$ a field. Is it possible to get back galois cohomology groups $H^*(k,G)$ studying the cohomology of the classifying stack $BG=[*/G]$ ?
1
vote
0answers
48 views

$\Gamma$ cohomology of principal series

Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup with Langlands ...
2
votes
1answer
113 views

rational cohomology of finite real grassmannian

Let $p_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E_n(\mathbb{R}^\infty)\to G_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, The Cohomology of BSO n and BO n with ...
2
votes
1answer
160 views

cohomology ring of configuration spaces

In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use ...
1
vote
1answer
133 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 ...
0
votes
0answers
49 views

Wedge product of Endomorphism-Valued Forms

To define characteristic classes in smooth vector bundles $E\longrightarrow M$ there is a more or less standard procedure: to choose a connection $\nabla$ and to derive the curvature $\Omega$, which ...
0
votes
1answer
92 views

Rationally graded singular cohomology

In the paper "A new cohomology theory for orbifold" by Chen/Ruan, they define the orbifold cohomology group of an orbifold $X$ by $H^d_{orb}(X)=\bigoplus_{(g) \in T} H^{d-2\iota_{(g)}}(X_{(g)})$ ...
0
votes
0answers
56 views

Crossed module classification theorem and $H^3(G; A)$

We are interested in "Crossed Module Classification Theorem (Two crossed modules with kernel $A$ and cokernel $G$ determined the same class in $H^3(G;A)$ if and only if they are equivalent.)". We ...
0
votes
0answers
50 views

question about Computations of gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces

In the paper Computations of gelfand-fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, F. R. Cohen, L. R. Taylor, Geometric Applications of Homotopy ...
0
votes
0answers
64 views

Dyer-Lashof algebra structures over graded modules

In Lecture Notes in Mathematics, Vol. 533, The homology of iterated loop spaces, Chapter 3, The homology of $C_{n+1}$-spaces, F. Cohen, Section 2, page 222, line 4, 5, 6: for an arbitrary graded ...
1
vote
1answer
115 views

Degree of irreducible locally free sheaves and global sections on curves

Let $X$ be a smooth projective curve and $\mathcal{F}$ a locally free sheaf on $X$ of rank $2$ and negative degree. Assume further that $\mathcal{F}$ is irreducible in the sense, $\mathcal{F}$ cannot ...
2
votes
1answer
97 views

cohomology algebra of submanifold in euclidean space

If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline} F(S^2,k+1)=\{(x_1,x_2,x_3,\cdots, ...
0
votes
0answers
89 views

permutation action on cohomology of configuration space

Let $F(M,n)$ be the $n$-th configuration (ordered) of manifold $M$. In the paper The Integral Cohomology Algebras of Ordered Configuration Spaces of Spheres, ...
1
vote
1answer
171 views

permutation action on cohomology of Stiefel manifolds

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds. In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ...
1
vote
4answers
264 views

Global section of very ample line bundles and its value on stalks

Let $X$ be a projective scheme and $\mathcal{L}$ be a very ample line bundle on $X$ with respect to some projective embedding $X \hookrightarrow \mathbb{P}^n_{\mathbb{C}}$ (for some $n$). Given any ...
4
votes
0answers
177 views

For which quiver varieties is Kirwan surjectivity known?

The cohomology of Nakajima quiver varieties is a quite interesting object. It's equipped with some natural classes given by the Chern classes of the tautological bundles associated to the spaces in ...
1
vote
0answers
172 views

Multiplication Map, Is it invariant?

Let $\pi:X\rightarrow Z$ a double cover of an elliptic curve with genus $g\geq 3$. Choose a general rank 2 and degree -1 vector bundle $F$ on $Z$, let $E=\pi^*F$ and fix $x\in X$. The involution $i$ ...
1
vote
1answer
112 views

configuration spaces of real projective space

Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring $$ H^*(F(\mathbb{R}P^n,k);R)$$ is obtained for any ...
9
votes
2answers
387 views

Is there an analogue of the Tate (and Hodge) conjecture for varieties that are not proper smooth (i.e., the mixed case)?

Let $X/K$ be a variety (scheme of finite type, geometricaly integral) over a finitely generated field $K$. If it is smooth and proper, we can formulate the Tate conjecture, and if $\text{char}(K) = 0$ ...
0
votes
0answers
120 views

Torsion in cohomology

Suppose to have a short exact sequence of chain complexes of $\mathbb{Z}$-modules: $$0\to A^\bullet\to B^\bullet\to C^\bullet\to 0$$ such that $A^k,B^k,C^k$ are non zero for $k=0,1,2$. Moreover, ...
7
votes
2answers
298 views

Deformations of a blowup

Let $S$ be a smooth projective surface over $\mathbb{C}$. (I guess this can be more general—higher dimension, other ground fields, non-projective, maybe even singular?—and I'dd like to hear that.) Let ...
0
votes
1answer
350 views

Axioms for sheaf cohomology

Let $R$ be a commutative ring and $X$ a topological space. Define a sheafy cohomology theory (see here) to be a collection of functors $\mathrm{H}^q:\mathrm{Sh}(X;R\mathrm{Mod})\to R\mathrm{Mod}$ such ...
1
vote
0answers
189 views

Intuitive Approach to Sheaf and Cech Cohomology [closed]

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
5
votes
2answers
347 views

Applications of cohomology to probability and statistics

Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory? By "interesting/useful", I mean "not merely ...
2
votes
1answer
122 views

cohomology of orthogonal (or general linear) group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $$ O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\} $$ What is $$ ...
6
votes
2answers
346 views

Hard Lefschetz Theorem for the Flag Manifolds

In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See herehere for a description.) I would like to know what the hard Lefschetz ...
1
vote
1answer
184 views

cohomology of orthogonal group of integers

Let $$ O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k). $$ What is $$ H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})? $$ If it cannot be computed out, can we get $$ H^*(O(\mathbb{Z}^{\oplus ...
2
votes
1answer
302 views

cohomology ring of symmetric group of order $3$

Let $S_3$ be the symmetric group of order $3$. What is the cohomology ring $$ H^*(S_3;\mathbb{Z})?$$ My attempt: I want to use mathematical induction on $n$ for $S_n$. For $n=1$, $S_1$ is trivial. ...
8
votes
1answer
334 views

Automorphisms of generic complete intersections

This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions. To state the question, let $r \geq ...
1
vote
0answers
75 views

Dixon's diagram for BRS cohomology

The article by J. A. Dixon titled Calculation of BRS cohomology with spectral sequences (Comm. Math. Phys. Volume 139, Number 3 (1991), pages 495-526) describes in words a diagram that is not printed. ...
0
votes
2answers
270 views

integral or rational cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
3
votes
1answer
140 views

homology of configuration spaces of non-compact manifolds

Let $M$ be a manifold. Let $F(M,n)$ be the configuration space of $n$-tuples on $M$. Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered ...
1
vote
0answers
52 views

A cohomology associated with a codimension one foliation(2)

What is an example of a codimension one foliation of a manifold for which this cohomology is finite dimension for all dimension $*$? Moreover what is the description of this cohomology for ...
1
vote
1answer
85 views

A cohomology associated with a codimension one foliation

Let $\alpha$ be a non vanishing one form on a manifold which which defines a codimension one foliation. With this $\alpha$ we define the following complex: $$\phi:\Omega^{i}(M)\to ...
6
votes
2answers
130 views

An example of two cofibrant dg categories whose tensor product is not cofibrant

I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and ...
1
vote
0answers
96 views

Dividing a n- cochain by a 1-cochain

Assume that $X$ is a path connected hausdorff topological space. Let $\alpha\in C^{1}(X,\mathbb{R})$ and $\beta\in C^{n}(X,\mathbb{R})$ be cochains in real singular cohomology. Asume that $\alpha ...
9
votes
0answers
271 views

Who first talked about “holes” in homology?

The question Why do the homology groups capture holes in a space better than the homotopy groups? and many others here use the idea that homology counts the ``holes'' in a space. The comments on this ...
7
votes
1answer
321 views

Intuitive Aproach to Dolbeault Cohomology [closed]

(Duplicated from math.stackexchange) I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. ...
3
votes
0answers
58 views

Computing Dolbeault cohomology of some simple domains

I have seen computations of the Dolbeault cohomology groups on compact Kahler manifolds using Hodge theory. I have never seen the computation of Dolbeault cohomology for simple domains in ...