and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
The fibration lemma (Bousfield-Kan 72, Chapter II) in (parametrized) rational homotopy theory states sufficient conditions under which rationalization preserves homotopy fibers.
(rational fibration lemma)
Let
be a Serre fibration of connected topological spaces, with fiber $F$ (over any base point) also connected.
If in addition
the fundamental group $\pi_1(B)$ acts nilpotently on the homology groups $H_\bullet(F,k)$
(e.g. if $B$ is simply connected, or if the fibration is a principal bundle);
at least one of $A$, $F$ has rational finite type
then the cofiber of any relative Sullivan model for $p$ is a Sullivan model for $F$.
(Félix-Halperin-Thomas 00, Theorem 15.3, following Halperin 83, Section 16)
Moreover, if $CE(\mathfrak{l}B)$ is a minimal Sullivan model for $B$, then the cofiber of the corresponding minimal relative Sullivan model for $p$ is the minimal Sullivan model $CE(\mathfrak{l}F)$ for $F$:
(Félix-Halperin-Thomas 00, Corollary on p. 199)
But this cofiber, being the cofiber of a relative Sullivan model and hence of a cofibration in the projective model structure on dgc-algebras, is in fact the homotopy cofiber, and hence is a model for the homotopy fiber of the rationalized fibration.
Therefore (1) implies that:
On fibrations of connected rational finite type-spaces, where the fundamental group of the base space acts nilpotently on the homology groups of the fiber: rationalization preserves homotopy fibers.
(This is the fibration lemma orginally due to Bousfield-Kan 72, Chapter II.)
Aldridge Bousfield, Daniel Kan, Chapter II “Fiber Lemmas” of: Homotopy Limits, Completions and Localizations, Springer 1972 (doi:10.1007/978-3-540-38117-4)
Yves Félix, Stephen Halperin, Jean-Claude Thomas, Sections 14 and 15 of: Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (doi:10.1007/978-1-4613-0105-9)
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