**2**

votes

**0**answers

199 views

### generalized Atiyah-Hirzebruch spectral sequence from Postnikov truncation

The Atiyah-Hirzebruch spectral sequence
\begin{equation*}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E),\end{equation*}
computes the generalized homology $h$ of a total space $E$ of a Serre fibration ...

**25**

votes

**1**answer

784 views

### Making $\mathbb{Q}$-cohomology integral

Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$
with $i>0$.
Does there always exist a variety $Y$ and a ...

**7**

votes

**1**answer

309 views

### cohomology of classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$.
Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$.
Let $\rho: ...

**14**

votes

**5**answers

695 views

### Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?

First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} ...

**5**

votes

**1**answer

377 views

### Fundamental class in K-theory and orientability

In ordinary homology, the classical results give the following situation:
for a compact, connected, topological manifold $M$ of dimension $n$ we have, for each ring $R$, that $H_n(M,M \setminus ...

**1**

vote

**2**answers

200 views

### The cohomology groups of $\Omega U(n)$

Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?

**2**

votes

**0**answers

230 views

### l-adic cohomology and perverse sheaves

Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...

**1**

vote

**0**answers

57 views

### Which groups may be obtained as $K$-homology groups?

Recently I asked the following question, about the separability of the underlying $C^*$-algebra in the definition of $K$-homology:
mathoverflow.net/questions/181361
As far as I understood, ...

**1**

vote

**0**answers

188 views

### Finite Cohomology and free groups

Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H ...

**5**

votes

**0**answers

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 ...

**43**

votes

**2**answers

1k views

### $H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
...

**2**

votes

**0**answers

228 views

### 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 ...

**4**

votes

**2**answers

388 views

### Cohomology of the toric variety $X_\Sigma=\mathbb C^2\sqcup \mathbb C^2\big/\left((x,y)_1\sim(x^{-1},y^{-1})_2\right)_{x,y\neq 0}$

I'm writing a thesis on Chow rings of toric varieties and am looking for a reference on the singular cohomology ring of the blowup of $\mathbb C^2$ at xy=0, i.e. at the coordinate axes. The ...

**1**

vote

**0**answers

103 views

### Differences and relationships between Motivic cohomology and Universal cohomology theory? [duplicate]

Differences and relationships between Motivic cohomology (Beilinson, Lichtenbaum and Voevodsky) and Universal cohomology theory (Grothendieck)?

**5**

votes

**1**answer

268 views

### Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$:
$0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...

**3**

votes

**0**answers

140 views

### Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles.
...

**0**

votes

**1**answer

108 views

### Cohomology of a fibered surface

Let $R$ be a complete Henselian discrete valuation ring, $\pi:X \to \mathrm{Spec} (R)$ be a smooth, proper, integral, flat $\mathrm{Spec} (R)$-scheme of dimension $2$. Assume that the genus of the ...

**4**

votes

**0**answers

129 views

### Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...

**5**

votes

**1**answer

214 views

### What properties of line bundles can be detected cohomologically?

Let $X$ be a proper, finite type scheme over a field $k$. What useful properties of line bundles (e.g. amplitude, nefness) can be detected cohomologially?
For example, in our setting we have the ...

**3**

votes

**0**answers

206 views

### extension problem for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...

**4**

votes

**2**answers

332 views

### stability results for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...

**7**

votes

**2**answers

746 views

### What morphisms induce injective/surjective maps on (Weil) cohomology?

Let $k$ be a field, let $f \colon X \to Y$ be a morphism of $k$-varieties, and assume $X$ and $Y$ are smooth and projective. Let $H(\_)$ be a classical Weil cohomology theory (i.e. one of $\ell$-adic ...

**0**

votes

**1**answer

170 views

### help with cohomology of $\mathbb{P}^n$ relative to a NCD

Let $H_0, \ldots, H_n$ be $n$ hyperplanes in $\mathbb{P}^n(\mathbb{C})$ with normal crossings and denote by $H$ the union of them. I am trying to understand why
(1) $H^n(\mathbb{P}^n(\mathbb{C}), ...

**7**

votes

**2**answers

293 views

### A cohomology group which depends on the connection

Warning: I am not a differential geometer, so some of the following might not make sense.
Background:
Let $w: (T\Omega)^k \to \mathbb{R}$ be a $k$-tensor on $\Omega$, an open subset of ...

**4**

votes

**1**answer

181 views

### Interpretations of differentials in hypercohomology spectral sequences as Yoneda products

I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups.
More ...

**7**

votes

**0**answers

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 ...

**5**

votes

**0**answers

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 ...

**9**

votes

**1**answer

323 views

### Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...

**3**

votes

**1**answer

144 views

### Can André–Quillen homology detect the property of being Gorenstein?

Let $(A,m,k)$ be commutative noetherian local ring.
Can one detect if $A$ is a Gorenstein ring from the André–Quillen homologies $H_n(A,k,-)$?

**11**

votes

**1**answer

268 views

### A cdga for compactly supported cohomology (à la Sullivan's algebra of polynomial forms)

Let $M$ be a smooth manifold, and let $\Omega^\bullet(M)$ be the commutative dg-algebra of differential forms on $M$. It is quasi-isomorphic to the dg-algebra of singular cochains on $M$. If $M$ is no ...

**4**

votes

**2**answers

298 views

### Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...

**12**

votes

**0**answers

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 ...

**20**

votes

**0**answers

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 $n \geq 3$ ...

**1**

vote

**0**answers

248 views

### Functors with Mayer-Vietoris Sequences

Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...

**2**

votes

**0**answers

104 views

### $D^{\infty}$ modules on analytic spaces

In Mebkhout's paper on Local Cohomology of Analytic Spaces, the following theorem is stated:
Let $X$ be a complex smooth manifold and $Y$ is an analytic subspace of $X$. Then ...

**1**

vote

**0**answers

141 views

### (Co)homology of classifying space of spin group $BSpin(n)$

In the answer for question: Homology of classifying space of spin group BSpin(n),
it was shown that $H_i(BSpin(\infty),Z)$ is $0,0,0,Z$, for $i=1,2,3,4$.
What is $H_i(BSpin(\infty),Z)$ or ...

**15**

votes

**1**answer

367 views

### Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google:
Suppose I have a countable field, $k$. ...

**5**

votes

**1**answer

491 views

### When does a cohomology class induce an isomorphism between homotopy groups?

A representative $\alpha$ of a cohomology class $[\alpha]\in H^n(X;\pi_n(X))$ is equivalent to a map from $X$ into an Eilenberg-Mac Lane space as $\alpha:X\to K(\pi_n(X),n)$. After applying a ...

**10**

votes

**0**answers

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 ...

**1**

vote

**1**answer

170 views

### What are finite groups $H$ such that $H^n(H,\mathbb{Q/Z}) \cong H_n(G,\mathbb{Z})$?

Let $G$ be a finite group and $G^{\prime}$ be its commutator subgroup. Let $\mathbb{Z}$ and $\mathbb{Q}$ denote the integers and rationals. $\mathbb{Z}$ and $\mathbb{Q/Z}$ treated as trivial ...

**3**

votes

**1**answer

323 views

### Compactly supported cohomology of homotopy equivalent manifolds

Are there examples of homotopy equivalent smooth, orientable manifolds $M$ and $N$ of the same dimension with non-isomorphic compactly supported cohomology rings?

**4**

votes

**0**answers

228 views

### Cohomology of BG, algebraically

Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient ...

**4**

votes

**1**answer

274 views

### Group extensions isomorphic as groups

Let $G$ be a group and $A$ a $G$-module. It well know that there is a group isomorphism between the second cohomologoy group $H^2(G,A)$ and the abelian group $OpExt(G,A)$ of classes of extension ...

**12**

votes

**1**answer

764 views

### Are there any cool applications of the generalized Atiyah-Hirzebruch(-Serre) spectral sequence?

Both the Atiyah-Hirzebruch and the Serre spectral sequence can be constructed from a skeletal filtration of a CW-complex:
We can construct the Atiyah-Hirzebruch spectral sequence by filtering $X$ by ...

**1**

vote

**2**answers

241 views

### A cohomology associated to a 1- form

In this question all objects are real analytic.(manifolds, differential forms..)
Assume that $M$ is a compact manifold and $\alpha \in \Omega^{1}(M)$ is a one form.
We define a map ...

**2**

votes

**1**answer

257 views

### mixed Hodge structure on Cohomology with compact support

Let $X$ be a smooth quasi-projective variety of dimension $n$ over the complex numbers (let us assume it is the complement of a normal crossings divisor in a smooth projective variety).
Then how does ...

**8**

votes

**1**answer

373 views

### worked out examples in borel-moore homology

I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking ...

**11**

votes

**2**answers

519 views

### Postnikov's algebraic reconstruction of cohomology from homotopy invariants

In his short paper (1951) and longer monograph (1955), Postnikov introduced what I believe are now called Postnikov systems or towers. It is my understanding that Postnikov systems have since then ...

**5**

votes

**2**answers

231 views

### The Borel construction of equivariant cobordism

In $K$ theory, the Borel construction of equivariant cohomology is somehow not the right one. The $G$-equivaraint $K$ theory of a point should be the representation ring of $G$, but $K(BG)$ is this ...

**4**

votes

**2**answers

1k views

### Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has
$$\otimes: D^{-}(X) \times D^{-}(X) \to ...