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Results tagged with cohomology
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user 318
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, ...
31
votes
Does a "Chern character" exist for any generalized cohomology theory?
For any (connective) spectrum $E$ one may rationalise it to get a rational spectrum $E_\mathbb{Q}$, and a map $E \to E_\mathbb{Q}$. Now rational spectra split as wedges of Eilenberg-Mac Lane spectra, …
- 18.2k
15
votes
Accepted
Group Extensions and Line Bundles on $BG$
If $L \to BG$ is the complex line bundle, take the unit sphere bundle $S^1 \to S(L) \to BG$ and take $\pi_1$.
- 18.2k
11
votes
Accepted
Relation between cohomology of ordered and unordered configuration spaces?
If $F_n(M)$ denotes the ordered configuration space and $C_n(M)$ the unordered configuration space, the quotient map gives a map
$$H^*(C_n(M)) \longrightarrow H^*(F_n(M)).$$
If one takes rational coef …
- 18.2k
7
votes
Accepted
Is there a kind of Poincare duality for Borel equivariant cohomology?
With the cup product in group cohomology this yields maps
$$C^n(G ; C^p(M ; A)) \otimes C^m(G ;C_q(M, \mathbb{Z})) \to C^{n+m}(G ; C^p(M ; A) \otimes C_q(M, \mathbb{Z})) \to C^{n+m}(G;C_{q-p}(M;A))$$
and …
- 18.2k
3
votes
Accepted
$G$-action on the integral homology of a compact surface
We consider the Serre spectral sequence for the fibration sequence
$$S \to S/P \to BP$$
in $\mathbb{F}_p$-cohomology, which has the form $E_2^{p,q} = H^p(P ; H^q(S;\mathbb{F}_p)) \Rightarrow H^{p+q}(S/ …
- 18.2k
3
votes
Accepted
Pontryagin square on $Y\times S^1$ where $Y$ is three-dimensional
otimes \bar{\mathfrak{P}}(n)$$
where $\bar{\mathfrak{P}}(-)$ is the Postnikov square (i.e. the operation given on cochains by $u \mapsto u \cup \delta u$), and $\beta$ is the Bockstein to $\mathbb{Z}/4$-cohomology …
- 18.2k
3
votes
Accepted
A question on composites of pushforward and pullback
A cohomology class $x$ supported on $\{e\} \times Y$ is sent by $\pi^* \circ \pi_! …
- 18.2k
1
vote
Accepted
Concrete pull-back calculation along H-space map
I will assume that by $\wedge$ you meant $\times$, and did not mean to write reduced cohomology (because I don't think the $H$-space structure gives you a map out of the smash product, and $b_k \otimes …
- 18.2k