The cohomology tag has no usage guidance.

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77 views

### Has this special functor a left and a right adjoint?

I would like to know if there exists the left and the right adjoint functors of the functor : $ X \to \displaystyle \bigoplus_{ n \geq 0 } H^n ( X , \mathbb{Q} ) = \displaystyle \bigoplus_{ n \geq 0 } ...

**4**

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**0**answers

91 views

### the first chern class of complex vector bundles

Let $\xi^\mathbb{C}$ be a complex vector bundle over a manifold $M$ (or $CW$-complex $B$).
Case~1: $\xi^\mathbb{C}$ is a complex line bundle. Then the first Chern class
$c_1(\xi^\mathbb{C})$ is zero ...

**4**

votes

**1**answer

123 views

### mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...

**3**

votes

**1**answer

137 views

### Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this.
I wonder if there is any source telling about a ...

**9**

votes

**1**answer

262 views

### positions of a methane molecule with carbon atom at the origin

Let $\text{CH}_4$ be the molecule of Methane:
The four hydrogen atoms form vertices of a regular tetrahedron with the carbon atom in the center of the regular tetrahedron.
Here we regard all atoms ...

**3**

votes

**0**answers

130 views

### What is the formula for the homology class represented by the diagonal?

Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis
for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion).
Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...

**9**

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**0**answers

135 views

### Stiefel-Whitney class of fibre bundles

Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...

**7**

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66 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, ...

**3**

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**0**answers

146 views

### Brauer group of a rational variety

This is a follow-up question to this question. There and here $X$ is a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. My question is:
...

**6**

votes

**3**answers

296 views

### Examples of Stiefel-Whitney classes of manifolds

Let $M$ by an compact, connected $n$-dimensional manifold without boundary.
Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, ...

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**0**answers

92 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 ...

**4**

votes

**1**answer

171 views

### Kunneth formula of Cartesian product modulo orders of coordinates

Let $X$ be a topological space and $F$ a field. Let the $n$-th permutation group $\Sigma_n$ act on
$$
\prod_n X
$$
by
$$
\sigma(x_1,\cdots,x_n)=(x_{\sigma(1)},\cdots,x_{\sigma(n)}), \sigma\in ...

**2**

votes

**2**answers

212 views

### Is the “inverse” (i.e., the “cohomological”) numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra “acceptable”? [closed]

I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider ...

**3**

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**0**answers

40 views

### group action on Stiefel manifolds [duplicate]

In the paper The cohomology rings of real Stiefel manifolds
with integer coefficients, it is stated that
Question:
Suppose the permutation group $\Sigma_k$ acts on $V_{n,k}$ by permuting the order ...

**2**

votes

**1**answer

94 views

### cohomology ring of the fundamental group of unordered configuration space

From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR
APPLICATIONS, p. 18, I find:
Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above ...

**2**

votes

**0**answers

49 views

### construction of grassmannian manifolds as collection of subspaces of Euclidean space

The grassmannian $G_k(\mathbb{R}^n)$ is the collection of all $k$-dimensional linear subspaces of $\mathbb{R}^n$ equipped with the quotient topology. The cohomology ring of $G_k(\mathbb{R}^n)$ has ...

**3**

votes

**0**answers

134 views

### Possible Betti numbers of smooth complex varieties

Given a smooth projective complex variety $X$ of dimension $n$, there are various restrictions on its sequence of Betti numbers $b_0, b_1, ..., b_{2n}$. Of course, $b_0=b_{2n}=1$ and $b_i=b_{2n-i}$ by ...

**2**

votes

**0**answers

109 views

### Serre spectral sequence to compute cup product of bundles

Could we use Serre spectral sequence to compute cup product $\cup$ for the fibration
$$
F\to E\to S^m?
$$
My supervisor said this is true and wrote
$$
H^*(S^m)\to H^*(E)\to H^*(F),\\
\alpha\to ...

**1**

vote

**0**answers

50 views

### cohomology ring of iterated loop-suspension of spheres

In the book The unstable Adams spectral sequence for free iterated loop spaces (http://www.ams.org/bookstore?fn=20&arg1=memoseries&ikey=MEMO-36-258), Corollary 3.14:
Question: given $n$, ...

**2**

votes

**1**answer

147 views

### generalized universal coefficient sequence

Take the familiar Universal Coefficient Theorem for ordinary homology with $\mathbb{Z}$-coefficients and ordinary cohomology with coefficients in some abelian group $A$:$$0\rightarrow ...

**7**

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**1**answer

126 views

### cohomology of iterated loop space on spheres

In the book The homology of iterated loop spaces, the homology Hopf algebra
(1)
$$
H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p)
$$
for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...

**13**

votes

**1**answer

358 views

### Integral cohomology ring of K(Z,3)

Computing the cohomology of Eilenberg Maclane spaces is a feasible but difficult problem in algebraic topology. The general answer is quite complicated (see the MO answer and the reference therein ...

**3**

votes

**1**answer

68 views

### Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer.
Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...

**5**

votes

**2**answers

245 views

### cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...

**2**

votes

**1**answer

171 views

### Schubert calculus and Pieri's formula

In the lecture notes Grassmannians: the first example of a moduli space. MIT Open Course Ware. page 7:
Are there any formal publications (books/papers) where I can find the formula?

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votes

**1**answer

178 views

### group completion theorem of homology as Hopf algebras

Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a ...

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79 views

### Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a ...

**7**

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**1**answer

158 views

### Steenrod operations on cohomology of grassmannians

Let $G_k(\mathbb{R}^n)$, $n\geq k$ and $G_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of ...

**0**

votes

**2**answers

259 views

### When is the Thom class the Poincare dual of the zero section?

As the title suggests, when is the Thom class the Poincare dual of the zero section? For starters, it's true for the normal bundle of an immersion...

**3**

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**1**answer

115 views

### Stokes-like Theorem for Dolbeault Operator

I have a simple question regarding complex geometry: is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$? For instance, suppose that $M$ is a closed complex manifold ...

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49 views

### cohomology ring of cross-section space of one-point compactification of tangent bundle

Let $M$ be an $m$-manifold whose cohomology is known. Let $TM$ be the tangent bundle of $M$ and $\xi$ be the fibre-wise one-point compactification of $TM$. Then $\xi$ is a $m$-sphere bundle over $M$. ...

**0**

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**0**answers

97 views

### reference for groupoid cohomology

In nLab (groupoid cohomology) says:
"Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types."
Are there references for ...

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**0**answers

156 views

### Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...

**7**

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**1**answer

422 views

### Do modular forms show up in the cohomology of moduli spaces of unmarked curves?

Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + ...

**3**

votes

**1**answer

250 views

### What is the (co-)homology of $K(\mathbb{R}_\delta,n)$?

To elaborate on the question from the title, $\mathbb{R}_\delta$ is the additive group of real numbers (without any topology) and $K(\mathbb{R}_\delta,n)$ is an Eilenberg-MacLane space. I would like ...

**1**

vote

**1**answer

119 views

### When the restriction of derived equivalence to a summand is a derived equivalence as well

I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian ...

**9**

votes

**1**answer

230 views

### Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$.
Is the set of $S$-isomorphism classes of $G$-torsors ...

**0**

votes

**0**answers

99 views

### Chern classes, vanishing of smooth sections or vanishing of holomorphic?

I have seen both definitions and this is getting me more and more confused.
Are Chern classes dual to the degeneracy cycles of smooth sections or holomorphic?
They can't be the same thing, can they?
...

**3**

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**0**answers

69 views

### cohomology ring of mapping spaces

In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring
$$
H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p)
$$
is obtained for any ...

**3**

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**0**answers

85 views

### Configuration spaces of positive and negative particles

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled ...

**1**

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**1**answer

50 views

### Group completion of labelled configuration space on Euclidean spaces

In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology
$$
\alpha_n: C(\mathbb{R}^n;X)\to ...

**1**

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58 views

### cohomology ring of unordered configuration space on Euclidean spaces

Let $F(\mathbb{R}^n,k)/\Sigma_k$ be the $k$-th unordered configuration space on $\mathbb{R}^n$. In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 226, it ...

**6**

votes

**3**answers

481 views

### Homological vs. cohomological dimension of a group/space

I have several related questions regarding homological vs. cohomological dimension of a space/group (this is not a duplicate of this).
The standard definition of the cohomological dimension $cd(X)$ ...

**1**

vote

**1**answer

171 views

### Geometric interpretation of Chern classes over flag manifolds

I would like to find that Chern classes of the tautological bundles over a flag manifold are dual to some cells in homology, analogously to what happens for the Grassmanian case. I have not been able ...

**4**

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**1**answer

244 views

### Künneth formula for Bredon cohomology theory

Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...

**1**

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**0**answers

159 views

### Functors similar to $H^i(\cdot)$

Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and ...

**3**

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**1**answer

118 views

### group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281:
Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...

**0**

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**1**answer

296 views

### On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ ...

**2**

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**2**answers

454 views

### What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$?

I would like to know the cohomology groups $\mathrm H^\bullet(\mathrm B^n(\mathbb Z/2);\mathbb Z/2)$. I assume that this is a standard computation, but I'm not sure where to look up the answer (and, ...

**2**

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**0**answers

359 views

### Differential and pre-differential of Jacobi identity

Let M be a manifold.
To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied?
That is a Lie algebra structure for which $[X,fY]=f[X,Y]$.
(For ...