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Results tagged with group-cohomology
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user 35575
In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.
2
votes
How to fit res map into a long exact sequence?
I think you're looking for the Hochschild-Serre spectral sequence. It's slightly more complicated than a long exact sequence, but you can extract the very concrete "inflation-restriction sequence" ou …
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9
votes
2
answers
483
views
Coboundary Representations for Trivial Cup Products
Suppose $G$ is a pro-$p$-group, $p$ odd, and $\mathbb{F}_p$ is given the trivial $G$-action. By skew-symmetry of the cup-product in degree 1, given $\chi\in H^1(G,\mathbb{F}_p)$, we have $\chi\cup\ch …
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5
votes
Accepted
Injectivity of Transfer (Verlagerung) map
I'm not sure how to answer the more philosophical question (it's likely you could encode enough of the axioms to force the purely group-theoretic version of the question to be true, but to ask whether …
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9
votes
Essential theorems in group (co)homology
Shapiro's lemma and dimension-shifting. Cohomology of cyclic groups and Herbrand quotients.
It would be helpful to know what you need to know group cohomology for.
If you have an interest in pr …
31
votes
3
answers
3k
views
Why aren't there more classifying spaces in number theory?
Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested in...) …
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9
votes
Geometric interpretation of the lower central series for the fundamental group?
A special case you might find informative: If $L$ is a link in $S^3$, then Chen-Milnor theory gives you a presentation of the link group $\pi=\pi_1(S^3-L)$ modulo some deeper terms of the the lower c …
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