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Results tagged with cohomology
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user 3473
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, ...
6
votes
Group cohomology version of Deligne-Beilinson cohomology
cohomology. … It is equivalent to the Segal-Mitchison smooth group cohomology.
References on general simplicial Deligne cohomology are:
Brylinski, J.-L.; McLaughlin, D. …
- 4,519
2
votes
Accepted
Low dimensional integral cohomology of $BPSO(4n)$
Gawedzki and I have investigated this question for all compact simple Lie groups using the descent of multiplicative bundle gerbes from simply connected covers to quotients by subgroups of the center: …
- 4,519
8
votes
Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$
If $X$ has a universal cover $\widetilde{X}$, then $\widetilde X$ is a prinicpal $\pi_1(X)$-bundle over $X$. Suppose
$$
A \to E \to \pi_1(X)
$$
is a central extension. Then, the corresponding class in …
- 4,519
3
votes
Accepted
Connection Transformation Formula; Degree 3 Cech Cohomology
The equality follows directly from the definition of a connection, and is independent of the context of lifting structure groups, or degree three cohomology. …
- 4,519
4
votes
Continuous cohomology of semi-simple Lie group
For $K=G$ in the compact case, this relative Lie algebra cohomology is identically zero.
A good place to look is Stasheff's "Continuous cohomology of groups and classifying spaces". …
- 4,519
5
votes
Group Extensions and Line Bundles on $BG$
Your line bundle $L$ over $BG$ can be seen as a $G$-equivariant line bundle over a point. That is, up to isomorphism, just a continuous group homomorphism $f:G \to \mathbb{C}^{\times}$. Try to lift $f …
- 4,519
14
votes
Why do gerbes live in H^2?
The point is that there is an isomorphism
$$
H^1(X,B\mathbb{C}^\times) = H^2(X,\mathbb{C}^\times),
$$
which explains the relation to the second cohomology group. … All this is very nicely explained in Gajer's Inventiones paper "Geometry of Deligne cohomology". …
- 4,519