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Results tagged with cohomology
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user 5792
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, ...
2
votes
Accepted
Borel's transgression theorem for spectral sequences
Mimura and Toda's statement in The Topology of Lie Groups, Theorem VII.2.9 (p. 378), requires less severe degree constraints. They start with a Serre fibration $F \to E \to B$ with $B$ simply-connecte …
- 2,995
9
votes
0
answers
258
views
Reference request: $H^* X$-module structure on the Mayer–Vietoris coboundary
Let $E^*$ a multiplicative cohomology theory. … \cap V) \wedge I\big) \cup V \simeq X.$
And this one is stated as Proposition 2.15 of Allen Hatcher's Vector Bundles and K-Theory (for complex topological K-theory):
Let $E^*$ be a multiplicative cohomology …
- 2,995
4
votes
0
answers
93
views
A dimension condition on the cohomology of a homogeneous space
The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. …
- 2,995
2
votes
cohomology ring of homogenous manifold
The third arrow is a fixed but arbitrary choice of lifting of this module of generators to a subspace of the cohomology ring, and the last map is induced by the inclusion $H_0 \hookrightarrow G$. … One can compute the cohomology algebra then as
$$\mathbb R[s] / (s^2) \otimes \Lambda [z],$$
where $z_3 - 2sz_2/3$ represents $z$. …
- 2,995
7
votes
1
answer
529
views
Naive equivariant transfer
Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_! …
- 2,995
5
votes
1
answer
145
views
Example request: seriously deficient homogeneous spaces
The quotient map $G \to G/K$ induces a ring map $H^*(G/K) \to H^*(G)$ in rational cohomology whose image is an exterior algebra $\Lambda \hat P$. …
- 2,995
7
votes
Cohomology ring of mapping torus
Here are some more partial observations, overlapping some of the others. The result won't be completely explicit, as there are too many case distinctions for a clean answer, but hopefully at least the …
- 2,995
9
votes
Is there any "deep" relation between the localization theorem of equivariant cohomology and ...
I have never checked this, but because $R(G)$ is Noetherian, I believe the localization theorem in equivariant cohomology at the level of $\mathbb Z/2$-graded rings then follows from the K-theoretic localization …
- 2,995