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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.
I have been wanting to edit this page for a while, but have never been able to understand a few things necessary to do so satisfactorily. The notion of twisting cochain that I’m familiar with is what might be called a “topological twisting cochain”, I suppose: the definition is that found in Toledo and Tong, i.e. a Maurer–Cartan element in a certain bicomplex. Even though a few citations are listed at the bottom of the current page that give this definition, it’s not mentioned in the actual body of the nLab page. I’d like to add it, but (and here is the thing that I don’t understand) I don’t know how it relates to the definition that is given. Does anybody know of a reference that shows that twisting cochains (à la Toledo–Tong) are twisting cochains (in this dg/bar construction sense)?
Something that is written down (in my thesis, at least) is that these “topological” twisting cochains are specific examples of twisted complexes (à la Bondal–Kapranov), so maybe this comes into the story somehow.
Original papers are from late 1950, like Brown 1959 or so in my memory. Smirnov’s book on simplicial and operadic methods in algebraic topology has both as well as some papers from about a decade ago of Kathryn Hess. I was considering a bicategorical version after a question of Jurčo, full treatment uses supercoherence of Jardine but we never wrote it, as one should study the universal example to have some application. That was a while ago, so I am talking fro vague memory.
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