Talk:Rational homotopy theory

Formal spaces

I have changed the definition of Formal spaces in a significant way. I believe that the way the page was before was confusing and wrong.

For a formal space, its cohomology algebra should be a model for it. The question is whether it has to be the minimal model (rather than just a cdga model). I believe that the answer is no -- for example, even spheres are formal, but their minimal model is not their cohomology. In fact, if a space's minimal model is its cohomology, it must be a free alternating algebra on some generators. (Every minimal model is a Sullivan model, and thus is a free alternating algebra. If we assume it is also the cohomology, the differential has to be 0 as well). The only such spaces are products of Eilenberg-Maclane spaces.

I am not an expert on this, though, but please enlighten me if this is a mistake.

199.168.74.114 (talk) 21:35, 22 November 2014 (UTC) Ilya.[reply]

Todo

Add additional motivation for rational homotopy theory
Cleanup notation and improve organization
https://www.math.ias.edu/~lurie/ThursdayFall2017/Lecture1-Overview.pdf has an excellent overview of the ideas of the subject, they should be integrated into this page — Preceding unsigned comment added by Wundzer (talk • contribs) 18:03, 10 June 2020 (UTC)[reply]

Undefined concept

The section Formal spaces begins with this passage:

"A commutative differential graded algebra A, again with $A^{0}=\mathbb {Q}$ , is called formal if A has a model with vanishing differential. This is equivalent to requiring that the cohomology algebra of A (viewed as a differential algebra with trivial differential) is a model for A (though it does not have to be the minimal model). Thus the rational homotopy type of a formal space is completely determined by its cohomology ring."

But the term "formal space" has never been defined in the article!

I hope someone knowledgeable about this subject will include the definition there in the article.

(The next part of the article provides several examples of formal spaces, all without the concept ever having been defined.)

"Topologically nilpotent" listed at Redirects for discussion

The redirect Topologically nilpotent has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2025 March 6 § Topologically nilpotent until a consensus is reached. 1234qwer 1234qwer 4 21:25, 6 March 2025 (UTC)[reply]