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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, ...
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
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
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
0
votes
1
answer
188
views
cohomology algebra of unordered configuration space on Euclidean space
In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents):
Line 2: ... is used to compute the precise algebra struct …
- 1,990
2
votes
1
answer
336
views
fiber sequence of principal bundles
Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle
$$
G\to E\to B,$$
then $B=E/G$, the orbit space under action of $G$.
Let $BG$ be the classifying space of $G$. …
- 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
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 $$
H^*(BO(\mathbb{Z}_2^{ …
- 1,990
9
votes
1
answer
540
views
cohomology of classifying space of permutation groups
Let $\rho^*: H^*(G_k(\mathbb{R}^\infty))\to H^*(B\Sigma_k)$ be the induced homomorphism of cohomology with coefficients in $\mathbb{Z}_2$.
What is the image $\text{Im} \rho^*$? …
- 1,990
4
votes
1
answer
600
views
Relation between cohomology of ordered and unordered configuration spaces
Suppose the cohomology algebra $H^*(F(M,p);\mathbb{Z}/p\mathbb{Z})$ is known. … $, the question is answered at Relation between cohomology of ordered and unordered configuration spaces? …
- 1,990
4
votes
1
answer
419
views
homology of configuration spaces of non-compact manifolds
Let $M$ be a manifold.
Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.
Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered config …
- 1,990
4
votes
1
answer
741
views
cohomology ring of symmetric group of order $3$
What is the cohomology ring
$$
H^*(S_3;\mathbb{Z})?$$
My attempt: I want to use mathematical induction on $n$ for $S_n$.
For $n=1$, $S_1$ is trivial. …
- 1,990
1
vote
1
answer
612
views
cohomology of orthogonal group of integers
Let
$$
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
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
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
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