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Results tagged with cohomology
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user 41075
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, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
5
votes
1
answer
404
views
triviality of homology with local coefficients
Let $X$ be a manifold or a CW-complex.
Let
$\pi: \tilde X\longrightarrow X$
be a covering map.
Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an ort …
- 1,990
2
votes
1
answer
270
views
can the actions of fundamental groups annihilate homology?
Let $X$ be a path-connected manifold (or a CW complex).
Let $\pi_1(X)$ be the fundamental group of $X$.
Let $\pi: \tilde X\longrightarrow X$ be a covering map.
For each $m\geq 0$, let $C_m(\tilde X)$ …
- 1,990
7
votes
0
answers
419
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 …
- 1,990
1
vote
0
answers
125
views
cohomology ring of compact submanifolds of Euclidean spaces
Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$. … Question: are there any software / programming that can give the cohomology ring $H^*(M)$ automatically for any compact submanifold of Euclidean spaces, given by finitely many equations of coordinates? …
- 1,990
1
vote
1
answer
297
views
torsion part of the cohomology module of configuration spaces of manifolds
I want to find the mod $p$ torsion part of the cohomology module
$$
H^*(C_p(M);\mathbb{Z})
$$
for any prime $p$. … If I cannot find a statement for general $M$, then I want to find as many as possible examples of $M$ such that the mod $p$ torsion part of the cohomology module
$
H^*(C_p(M);\mathbb{Z})
$
are known …
- 1,990
2
votes
1
answer
619
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?
- 1,990
-2
votes
1
answer
290
views
stable splitting into a wedge sum [closed]
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
$$
X\to \Sigma^tX\to \bigvee _{k=1}^\inf …
- 1,990
1
vote
1
answer
292
views
Unordered configuration space of $\mathbb{R}P^1$
In the paper
GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL
ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2):
159-165,
Theorem 2. (b): $TP^n(\mathbb{R}P^1)$ is ho …
- 1,990
3
votes
1
answer
364
views
cohomology ring of configuration spaces
In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. … How to obtain the cohomology ring $H^*(F(M,k);\mathbb{F})$, where $F(M,k)$ is the ordered configuration space, $\mathbb{F}$ is a field?
Where could I find the proof? …
- 1,990
3
votes
1
answer
1k
views
cohomology of the orbit space of a group action
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain $H^*(M/G;F)$. Is there any method or procedure to follow? …
- 1,990
2
votes
1
answer
129
views
cohomology algebra of submanifold in euclidean space
1,\\
\text{ for }i\neq j, x_{3i+1}\neq x_{3j+1} \text{ or } x_{3i+2}\neq x_{3j+2} \text{ or }x_{3i+3}\neq x_{3j+3} \},
\end{multline}
is there any computer software or programming that can give the cohomology … Can the computer give a very complicated simplicial complexes to approximate the manifold and compute the cohomology algebra? …
- 1,990
1
vote
1
answer
257
views
permutation action on cohomology of Stiefel manifolds
In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto Univ. … Volume 43, Number 2 (2003), 411-428.,
the cohomology rings
$$
H^*(V_k(\mathbb{R}^n);\mathbb{Z}_2),
$$
$$
H^*(V_k(\mathbb{R}^n);\mathbb{Z}),
$$
are obtained. …
- 1,990
1
vote
1
answer
373
views
configuration spaces of real projective space
In http://arxiv.org/abs/1502.04258, the cohomology ring
$$
H^*(F(\mathbb{R}P^n,k);R)$$
is obtained for any commutative ring $R$ with unit and $2$ invertible. …
- 1,990
2
votes
1
answer
385
views
cohomology of orthogonal (or general linear) group over finite fields
Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let
$$
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\}
$$
What is $$
H^*(BO(\mathbb{Z}_2^{ …
- 1,990
1
vote
1
answer
612
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 …
- 1,990