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.

added to *KK-theory* brief remark and reference to relation to stable $\infty$-categories / triangulated categories

I am giving *fiber integration in K-theory* a dedicated entry.

One section *In operator K-theory* used to be a subsection of *fiber integration in generalized cohomology*, and I copied it over.

Another section *In terms of bundles of Fredholm operators* I have now started to write.

Started an entry in “category:motivation” on *fiber bundles in physics*.

(prompted by this Physics.SE question)

]]>some basics at *Steenrod algebra*

at *vector bundle* I have spelled out the proof that for $X$ paracompact Hausdorff then the restrictions of vector bundles over $X \times [0,1]$ to $X \times \{0\}$ and $X \times \{1\}$ are isomorphic.

It’s just following Hatcher, but I wanted to give full detail to the argument of what is now this lemma.

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

I tried to brush-up the References at *period* a little.

I have trouble downloading the first one, which is

- M. Kontsevich, Don Zagier, Periods (pdf)

My system keeps telling me that the pdf behind this link is broken. Can anyone see it? (It may well just be my system misbehaving, wouldn’t be the first time…).

]]>I gave *chromatic homotopy theory* an Idea-section.

To be expanded eventually…

]]>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.)

]]>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...

]]>have created an entry for *Bott periodicity*

I need to point to *[[reduced homology]]*, so I created an entry. But nothing much there yet.

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

created *Hodge structure*. Currently with nothing but a pointer to this nice book:

- Chris Peters, Jozef Steenbrink,
*Mixed Hodge Structures*, Ergebisse der Mathematik (2007) (pdf)

Eventually I’d think we should move over Hodge-structure articles from *Hodge theory* to here. But not tonight.

I have split off spin^c from spin^c structure

]]>added to group extension a section on how group extensions are torsors and on how they are deloopings of principal 2-bundles, see group extension – torsors

]]>Wrote some minimum at *natural bundle*.

finally an Idea-section at *Poincaré duality*.

(Needs more work, clearly, but should be a start)

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

wrote out the definition *In complex geometry*

added pointer to

- Tom Lovering,
*Etale cohomology and Galois Representations*, 2012 (pdf)

for review of how Galois representations are arithmetic incarnations of local systems/flat connections. Added the same also to *local system* and maybe elsewhere.

stated a kind of Idea/definiton at *motivic Galois group*.

Experts and experts-to-be, please check!

]]>I’d be trying to write out a more detailed exposition of how fiber integration in twisted generalized cohomology/twisted Umkehr maps are axiomaized in linear homotopy-type theory.

To start with I produced a dictionary table, for inclusion in relevant entries:

]]>created Stokes theorem

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