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...

]]>created *equivariant de Rham cohomology* with a brief note on the Cartan model.

(I seem to remember that we had discussion of this in the general context of Lie algebroids elsewhere already, several years back. But now I cannot find it….)

]]>added to *equivariant K-theory* comments on the relation to the operator K-theory of crossed product algebras and to the ordinary K-theory of homotopy quotient spaces (Borel constructions). Also added a bunch of references.

(Also finally added references to Green and Julg at *Green-Julg theorem*).

This all deserves to be prettified further, but I have to quit now.

]]>started *Galois cohomology*

I have split off *complex projective space* from *projective space* and added some basic facts about its cohomology.

created [[motive]] just in order to link to the sub-pages on this that we already have, and in order to record a link to a useful MO discussion about them.

]]>started a category:reference entry

in the course of this I added some stuff here and there, for instance at *Abel-Jacobi map*. But very stubby for the moment.

created *red-shift conjecture*

I have added also the statement of the two relative versions of the Serre spectral sequence (here). No details yet.

]]>stub for *equivariant elliptic cohomology*, for the moment just to record the references given there

stub for *Atiyah-Segal completion theorem*, for the moment just to record a reference

added to *representation ring* a brief remark on its relation to the equivariant K-theory of the point (very stubby still)

added a tiny bit of basics to *complex oriented cohomology theory*

At the old entry *cohomotopy* used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.

(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

]]>created stub entry for *Walker-Wang model*

An old query at twisting cochain which was reaction to somebody putting that *the* motivation is the homological perturbation lemma:

It is equally true that it is related to 20 more areas like that one (which is not the central). Brown’s paper on twisting cochains, is much earlier than homological perturbation theory. basic idea was to give algebraic models for fibrations. Nowedays you have these things in deformation theory, A-infty, gluing of complexes on varieties, Grothendieck duality on complex manifolds (Toledo-Tong), rational homotopy theory etc. One should either give a fairly balanced view to all applications or not list anything, otherwise it is not fair. This should be done together with massive expansion of Maurer-Cartan equation what is almost the same topic. The same with literature: Smirnov’s book on simplicial and operadic methods in algebraic topology is the most wide reference for twisting cochains and related issues in algebraic topology setup; Keller wrote much and well about this and Lefèvre-Hasegawa thesis (pdf) is very good, and the first reference is E. Brown’s paper from 1959. For applications in deformation theory there are many references, pretty good one from dg point of view and using 2-categorical picture of def functors is a trilogy of Efimov, Lunts and orlov on the arXiv. Few days ago Sharygin wrote a long article on twisting cochains on the arXiv, with more specific purposes in index theory. Interesting is the application of Baranovsky on constructing universal enveloping of L infty algebra. – Zoran

Urs: concerning the “either give a fairly balanced view to all applications or not list anything”, I can see where you are coming from, Zoran, but I would still prefer here to have a little bit of material than to have none. The $n$Lb is imperfect almost everywhere, we’ll have to improve it incrementally as we find time, leisure and energy. But it’s good that you point out further aspects in a query box, so that we remember to fill them in later.

Zoran Skoda My experience is that correcting a rambling and unbalanced entries takes more time than writing a new one at a stage when you really work on it. Plus all the communication explaining to others who made original entry which is hastily written. When it becomes very random and biased I stopped enjoying it at all to work on it.

Ronnie Brown It may not possible for one person to give a “balanced entry” and is certainly not possible for me in this area. On the other hand, this may be endemic to the description of an area of maths for students and research workers.

An advantage of the Homological Perturbation Lemma (HPL) is that it is an explicit formula, and this has been exploited by various writers, especially Gugenheim, Larry Lambe and collaborators, Huebschmann, and others, for symbolic computations in homological algebra. It is good of course to have the wide breadth of applications of twisting cochains explained.

For me, an insight of the HPL was the explicit use of the *homotopies* in a deformation retract situation to lead to new results. This has been developed to calculate resolutions of groups, where one is constructing inductively a universal cover of a $K(G,1)$ with its contracting homotopy.

So let us continue to have various individually “unbalanced” points of view explained in this wiki, to let the readers be informed, and decide.

*Toby*: Knowing basically nothing about this, I prefer to see various people explain their own perspectives. Even if they don't try to take the work to integrate them.

have added to *associated bundle* an exposition, here, of how in a context of $(2,1)$-topos theory, associated bundles are naturally thought of as homotopy pullbacks of action groupoid projections.

This is from one subsection of what I am currently typing into *geometry of physics – representations and associated bundles*. It may need more polishing, but I need to interrupt now for half an hour or so.

quick note at *spin structure* on the characterization *over Kähler manifolds*

have created an entry for *Bott periodicity*

created an entry *twisted Umkehr map*. The material now has some overlap with what I just put into *Pontrjagin-Thom collapse map*. But that doesn’t hurt, I think.

added some lines to *differential algebraic K-theory*

also a stub *Beilinson regulator*

created *Baum-Connes conjecture* with an emphasis on the *Green-Julg theorem* (of the statement in KK-theory).

I have started one of those hyperlinked indices at he “reference”-entry *K-Theory for Operator Algebras*.

I gave the brane scan table a genuine $n$Lab incarnation and included it at *Green-Schwarz action functional* and at *brane*.

created *cobordism theory determining homology theory* with the basic references (any more results along these lines?), also added a brief cross-link paragraph at *Landweber exact functor theorem*.

I couldn’t think of a better title, suggestions are welcome.

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