The cohomology tag has no wiki summary.

**0**

votes

**0**answers

17 views

### Schubert Polynomials for Complex Projective Space

The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascaux and Sch\"utzenberger gave specific ...

**2**

votes

**2**answers

222 views

### Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$?

In the web page http://www.encyclopediaofmath.org/index.php/Moore_space it can be found the following statement:
If $K(\mathbb Z,n)$ is the Eilenberg–MacLane space of the group of integers ...

**1**

vote

**1**answer

131 views

### Cohomology of Homogeneous Complex Manifolds

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...

**-2**

votes

**1**answer

57 views

### stable splitting into a wedge sum [on hold]

Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum
$$
\Sigma^t X\cong \bigvee _{k=1}^\infty Y_k.
$$
(1). Does this imply
$$
X\to \Sigma^tX\to \bigvee ...

**5**

votes

**2**answers

226 views

### Seifert Fibrations and their associated Spectral Sequence

In a somewhat limited setting, a Seifert Fibre Space is a 3-manifold $M$ with a "nice" decomposition into circles (http://en.wikipedia.org/wiki/Seifert_fiber_space). That is, $M$ is decomposed into ...

**5**

votes

**2**answers

207 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 ...

**6**

votes

**4**answers

612 views

### Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?

That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...

**1**

vote

**1**answer

190 views

### What is the cokernel of the map $H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow H^2(X,\mathbb Z).$

For a manifold $X$ (for simplicity, assumed to be compact), let $\pi_1(X)$ be the fundamental group of $X$. What is the cokernel of the map $$H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow ...

**8**

votes

**2**answers

423 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 ...

**2**

votes

**0**answers

87 views

### Generating free groups by small subgroups and an element

Let $F$ be a free group of countably infinite rank, and let $L \leq F$ be a finite index subgroup. For some prime $p$ and $k \in \mathbb{N}$, let $\sigma \colon L \to ...

**3**

votes

**2**answers

196 views

### Upper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)

I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists.
Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let ...

**2**

votes

**1**answer

114 views

### Unordered configuration space of $\mathbb{R}P^1$

In the paper
GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL
ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2):
159-165,
Theorem 2. (b): ...

**1**

vote

**1**answer

174 views

### Multiplicative structure in the cohomological Leray-Serre spectral sequence - please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set:
$X_p = \pi^{-1}(B^p)$,
$F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...

**-2**

votes

**0**answers

69 views

### Cohomology of conic bundle 3-folds [migrated]

It is known that for a smooth cubic 3fold $X\subset \mathbb{P}^4$ we have $H^3(X,\mathcal{O}_X)$ (or if you prefer $H^{0,3}(X)=0$). Moreover, if I project off a line $l\subset X$ I can resolve the map ...

**17**

votes

**1**answer

493 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

**1**answer

307 views

### 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 ...

**3**

votes

**1**answer

163 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

**0**answers

93 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

**1**answer

107 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 ...

**9**

votes

**2**answers

1k views

### The De Rham Cohomology of $\mathbb{R}^n - \mathbb{S}^k$

I'm reading Madsen and Tornehave's "From Calculus to Cohomology" and tried to solve this interesting problem regarding knots.
Let $\Sigma\subset \mathbb{R}^n$ be homeomorphic to $\mathbb{S}^k$, show ...

**7**

votes

**1**answer

246 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

**0**answers

60 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

**1**answer

243 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 ...

**40**

votes

**1**answer

1k views

### “Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...

**3**

votes

**2**answers

857 views

### Non-zero sheaf cohomology

Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is ...

**2**

votes

**1**answer

159 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

**0**answers

81 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

**1**answer

152 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

**0**answers

49 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

**1**answer

118 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

**1**answer

162 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

**1**answer

140 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

**0**answers

55 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

**1**answer

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

**0**answers

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

**0**answers

53 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 ...

**3**

votes

**0**answers

257 views

### Equivariant and basic cohomology

I have difficulties to understand the connection between equivariant and basic cohomology. I understand the definition of them but not how they are related (the Weil algebra killed me :)).
For ...

**0**

votes

**0**answers

65 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

**1**answer

121 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 ...

**7**

votes

**1**answer

1k views

### De Rham homology

Suppose M is an arbitrary smooth manifold and D is its bundle of 1-densities.
On the category of finite-dimensional vector bundles over M and linear differential operators between them
there is a ...

**2**

votes

**1**answer

101 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, ...

**1**

vote

**1**answer

173 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 ...

**4**

votes

**0**answers

180 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

**4**answers

270 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 ...

**0**

votes

**0**answers

91 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, ...

**18**

votes

**0**answers

480 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 ...

**1**

vote

**0**answers

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 $i$ ...

**9**

votes

**2**answers

393 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 $\text{char}(K) = 0$ ...

**1**

vote

**1**answer

113 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 ...

**3**

votes

**0**answers

315 views

### Defining the cup product in Ext using a Kunneth formula

I want to make a Kunneth product of sorts on Ext. In particular, letting $C_*$ be a $R$-free resolution for $k$ over a $k$-hopf algebra $R$, elements in $Ext_R(k,k)$ are represented by maps in ...