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, ...

learn more… | top users | synonyms

41
votes
0answers
647 views

Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition

Consider the following theorem of Atiyah. Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if ...
20
votes
0answers
668 views

Calabi-Yau cohomology?

My question here is going to be this -- but I'll give a bit of background to explain myself in a moment: What has been done/what results are available on Calabi-Yau cohomology in degree $n \geq 3$ ...
20
votes
0answers
565 views

Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...
19
votes
0answers
624 views

What is classified by the (big) crystalline topos?

In his paper "Generic Galois Theory of Local Rings", G.C. Wraith states on p. 743 that the (big) crystalline topos "can be conveniently described in terms of the theory it describes". What exactly is ...
16
votes
0answers
282 views

Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps $$ B G \longrightarrow B \mathrm{GL}_1(A) $$ for $A$ an $E_\infty$-ring carrying an oriented ...
15
votes
0answers
488 views

Steenrod algebra at a prime power

Let $n=p^k$ be a prime power. When $k=1$, the algebra of stable operations in mod $p$ cohomology is the Steenrod algebra $\mathcal{A}_p$. It has a nice description in terms of generators and ...
15
votes
0answers
417 views

Cohomological characterization of CM curves

In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable ...
13
votes
0answers
439 views

To what extent does (co)homology of groups made discrete depend on set theory?

There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...
12
votes
0answers
210 views

K-theoretic version of Artin-Mazur formal groups?

An Artin-Mazur formal group is, when it exists, the deformation theory of ordinary cohomology of some degree, on some algebraic variety. My question here is: Has the generalization of the theory of ...
11
votes
0answers
190 views

Cohomology of a configuration space of points on $\mathbb C^\times$ with an additional restriction

Let $Conf_{1,n}^3$ be the configuration space of collections of $n$ distinct numbered points on the annulus $\mathbb C^\times$ with an imposed restriction: for any $r\in \mathbb R^+$ the circle ...
11
votes
0answers
357 views

Cohomological interpretations of quadratic form invariants over rings?

The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
11
votes
0answers
896 views

Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb ...
10
votes
0answers
202 views

“topological” Ochanine genus?

The Witten genus has famously been lifted to the string orientation of tmf ("topological Witten genus"). For the Ochanine genus I am aware of a lift to a "spin orientation of Tate K-theory", namely to ...
9
votes
0answers
303 views

Milnor-Stasheff Characteristic Classes Problem 7B, Borel 1953

There is the following Proposition 11.1 from Borel's 1953 paper La cohomologie mod 2 de certains espaces homogènes (see here). Proposition 11.1 The classes $w^i$ and $\overline{w}^j$ are related ...
9
votes
0answers
290 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 ...
9
votes
0answers
396 views

Cohomology and impossible figures

In connection with the MO question Occurrences of (co)homology in other disciplines and/or nature I recalled Roger Penrose's “On the cohomology of impossible figures": ...
9
votes
0answers
188 views

KK-theory by abelianized correspondences of smooth stacks?

Whith (Kasparov, bivariant) KK-theory I am left with the nagging feeling that the theory is "more fundamental", than has been made explicit, that there is a "more profound" universal characterization ...
9
votes
0answers
200 views

Is there a Hodge isomorphism theorem for part-tangential, part-normal, harmonic differential forms?

Let $M$ be an oriented compact Riemannian $n$-manifold with boundary $\partial M$. A differential $p$-form $\omega$ on $M$ is normal if $i^* \omega = 0$ holds, tangential if $i^* \star \omega = 0$ ...
9
votes
0answers
1k views

Is Witten's new method of quantization useful for geometric complexity theory?

The Kempf-Ness theorem (see e.g. arXiv:0912.1132) - that the algebraic quotient of geometric invariant theory is also a symplectic quotient - suggests (to me) that certain physical constructions used ...
8
votes
0answers
191 views

Evens norm as a derived functor

In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...
8
votes
0answers
842 views

Is there any “deep” relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex ...
7
votes
0answers
135 views

When is the projection of an induced fibration trivial on cohomology?

Let $p: E\to B$ be a fibration, and let $f: A\to B$ be a continuous map. In my applications, $E$ and $B$ are finite complexes, but $A$ need not be. Form the pullback $$ \begin{array}{ccc} W & \to ...
7
votes
0answers
899 views

Why is the Nil-Hecke Algebra appearing?

The Nil-Hecke algebra is defined to be the subalgebra of the endomorphism ring of $\mathbb{C}[x_1,\ldots,x_n]$ generated by the operators of multiplication by $x_i$ and the divided difference ...
7
votes
0answers
497 views

Integral decomposition of the diagonal (Chow motives)

Let $k$ be a field of characteristic zero and let $X$ be a smooth proper varity over $k$ of dimension $d$. The Künneth standard conjecture conjectures that there exist projectors $e_0, e_1, \ldots, ...
6
votes
0answers
108 views

What is the etale cohomology of a product?

Let $X$ be a (smooth, projective if you wish) variety over a field $k$. I'm mostly interested in $k=\mathbb{F}_q$. What can one say about the etale cohomology ring $H_{et}^*(X \times_k X)$, say for ...
6
votes
0answers
131 views

kernel of the mod $2$ Bockstein on the first cohomology group

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...
6
votes
0answers
143 views

Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and ...
6
votes
0answers
283 views

Real schubert calculus

Studying real enumerative questions I noticed, that the even-even Schubert varieties of the real Grassmannian $Gr:=Gr_{2k}(2n,\mathbb R)$ behave analogously to the complex case. I call a partition ...
6
votes
0answers
226 views

When is the cohomology cross product square nonzero?

Let $X$ be a finite CW complex, and $A$ an abelian group. Given a nonzero class in singular cohomology $$x\in H^k(X;A),$$ when is its cross product square $$x\times x\in H^{2k}(X\times X;A\otimes A)$$ ...
6
votes
0answers
365 views

Cohomology of a space with coefficients in a Lie group?

The following is a very naive construction, and I am almost embarrassed to ask questions about it. Suppose I have a connected simple graph $\Gamma = (\Gamma_0,\Gamma_1)$ (i.e., vertex set, edge set) ...
6
votes
0answers
392 views

Is singular cohomology representable by a (Voevodsky's) motivic complex?

For any $c>0$ does there exist an object $C$ of Voevodsky's $DM^{eff}_-$ (over the field of complex numbers) such that for any $i\le c$ and any smooth variety $X$ the $i$-th singular cohomology of ...
6
votes
0answers
465 views

Is there a direct way to compute the higher derived image sheaves of a family of $\mathbb{P}^n$s?

Let $V\rightarrow Y$ be a vector bundle of rank $n+1$ over $Y$, with $Y$ reasonably nice (I care about the case of smooth, irreducible affine). Let $X=\mathbb{P}(V)$ be the projectivization of $V$, so ...
5
votes
0answers
121 views

Two natural maps asssociated with the nerve of a cover

Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...
5
votes
0answers
92 views

cohomology ring of configuration spaces on $S^2$ and the projective plane

For a manifold $M$ and a positive integer $n$, the unordered configuration space $B(M,n)$ is the space consisting of all unordered collections of $n$ distinct points on $M$. Precisely, $$ ...
5
votes
0answers
117 views

Testing the vanishing of cohomology fiberwise for a proper morphism from an Artin stack

Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. ...
5
votes
0answers
89 views

configuration space of Riemannian manifolds with a parameter on the distance of distinct points

Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as $$ F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, ...
5
votes
0answers
106 views

cohomology ring of stable configuration spaces

Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...
5
votes
0answers
146 views

Are all these K3 surfaces supersingular?

Consider all the smooth K3 surfaces given by $X^4+W^2X^2+XW^3 = f(Y,Z,W)$ or $X^4+XW^3 = g(Y,Z,W)$ over $\mathbb F_{2}$ with $f$ or $g$ homogenous of degree 4. There are a lot of choices for $f$ and ...
5
votes
0answers
226 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 ...
5
votes
0answers
128 views

The action of Steenrod squares on the indeterminate in Bullett-Macdonald identities

The Bullet-Macdonald identity (c.f. On the Adem relations)is the following: $$P(s^2+st)P(t^2)=P(t^2+st)P(s^2).$$ where we have $P(s)=\Sigma _s s^iSq^i$. This is known to be equivalent to the Adem ...
5
votes
0answers
206 views

Which ring spectra have some kind of exponential map turning addition into multiplication?

This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time. What will follow is sort of vernacular but whether it can be ...
5
votes
0answers
198 views

Which pure categories related with Weil cohomology theories are semi-simple?

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...
5
votes
0answers
321 views

Generalized Hodge conjecture for cohomology of smooth non-proper varieties?

Is there such a statement known i.e. does there exist a conjectural description of the coniveau filtration for singular cohomology of a smooth non-proper variety over the field of complex numbers (in ...
5
votes
0answers
430 views

What is the right notion of equivariant Cech cohomology?

What is the right definition of equivariant Cech cohomology is so that given a $G$-space $X$, $H^1_G(X;H)$ classifies $G$-equivariant principal $H$-bundles on $X$?
4
votes
0answers
65 views

Relation of BRST model of equivariant cohomology and BRST cohomology?

I'm now reading Kalkman's paper "BRST model for equivariant cohomology and representation for equivariant Thom class". And I've seen his definition for BRST model is $B=W(\mathfrak{g})\otimes ...
4
votes
0answers
49 views

Reference request: local cohomology in disjoint union

I have a topological space $X$ and two disjoint, closed subspaces $Y$ and $Z$ of $X$. I believe that in this situation, for any abelian sheaf $\mathcal{F}$ on $X$ and any $p \in \mathbb{N}$, there is ...
4
votes
0answers
47 views

Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher

It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a ...
4
votes
0answers
79 views

Example request: seriously deficient homogeneous spaces

In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...
4
votes
0answers
64 views

A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...
4
votes
0answers
137 views

References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms $\langle \cdot , \cdot \rangle_i : C_i \times C_i \to ...