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added to group cohomology
in the section structured group cohomology some remarks about how to correctly define Lie group cohomology and topological group cohomology etc. and how not to
in the section Lie group cohiomology a derivation of how from the right oo-categorical definition one finds after some unwinding the correct definition as given in the article by Brylinski cited there.
it's late here and I am now in a bit of a hurry to call it quits, so the proof I give there may need a bit polishing. I'll take care of that later...
two more propositions at group cohomology Lie and topological group cohomology and two more references.
In reply to this MO question I have added to group cohomology a bunch of references for the group cohomology of various topological groups.
Promted by discussion in another thread I noticed that group cohomology still only had an Idea-section, no Definition-section.
It still really hasn’t, but I started adding something.
So I jotted down the definitions
It is fun to notice that the syntax in homotopy type theory
$\prod_{x \colon \mathbf{B}G} (* \to A)$is so very close to that in homological algebra
$Ext_{\mathbb{Z}G}(\mathbb{Z}, A )$(but of course vastly more general).
I quit now. Not because I mean to suggest that what I added is done in any way, but because I have to quit. I’ll try to expand on this and turn it into something actually readable tomorrow.
Note to self that we might add some words about Galois cohomology.
Todd: Perhaps Galois cohomology should be a separate entry with a link from group cohomology. (I looked at the Wikipedia entry on Galois cohomology…. ! )
I have been further working on the entry group cohomology.
I am still not really happy with it, but I think now it is at least taking shape.
I have entirely rewritten the Idea-section (same idea, but nevertheless rewritten) and tried to streamline various things following it.
The main thing missing now, to my mind, is more details that unwind the abstract definitions, beyond the case of dgree-2 that is already spelled out in some detail. Parts of this I am going to import from what is currently at projective resolution, parts of it still need to be written.
tried to clean up what is currently the section Simplicial constructions by giving it a more systematic organization.
I have expanded the section Degree-2 group cohomology (which spells out the explicit component formulas) and added a discussion of how every gorup 2-cocycle is cohomologous to a normalized one (scroll down a bit to see this).
added to the Definition-section at group cohomology remarks on the relation to the homotopy-version/derived functor of the invariants functor
(very brief remarks though. This whole section needs to be exapanded, eventually)
added pointer to Cadek 99 where the cohomology of $B O(n)$ with twisted integer coefficients is given. (Happened to need that.)
I have adjusted a little bit the formatting of the maps of tetrahedra at group cohomology – in degree 2 (if anyone recognizes any tetrahedra there…). Of course it still looks pitiful, but a shade of pitifulness less than it did before.
Added:
S. Eilenberg, S. MacLane, Relations between Homology and Homotopy Groups. Proceedings of the National Academy of Sciences 29:5 (1943), 155–158. doi.
Samuel Eilenberg, Saunders MacLane, Relations Between Homology and Homotopy Groups of Spaces. The Annals of Mathematics 46:3 (1945), 480. doi.
Samuel Eilenberg, Saunders MacLane, Relations Between Homology and Homotopy Groups of Spaces. II. The Annals of Mathematics 51:3 (1950), 514. doi.
Dmitri Faddeev, On factor-systems in Abelian groups with operators. (Russian), Doklady Akad. Nauk SSSR (N. S.) 58, (1947). 361–364.
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