nLab rational parametrized homotopy theory

Idea

Rational parametrized homotopy theory is parametrized homotopy theory in the approximation of rational homotopy theory.

Properties

Proposition

(rational fibration lemma)

Let

$$\array{ F &\overset{fib(p)}{\longrightarrow}& A \\ && \big\downarrow{}^{\mathrlap{p}} \\ && B }$$

be a Serre fibration of connected topological spaces, with fiber $F$ (over any base point) also connected.

If in addition

1. 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);

2. 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 $F$:

(1)$$\array{ CE(\mathfrak{l}F) &\overset{ cofib \big( CE(\mathfrak{l}p) \big) }{\longleftarrow}& CE(\mathfrak{l}_{{}_B}A) \\ && \big\uparrow{}^{\mathrlap{ CE(\mathfrak{l}p) }} \\ && CE(\mathfrak{l}B) }$$

(Félix-Halperin-Thomas 00, Corollary on p. 199, Felix-Halperin-Thomas 15, Theorem 5.1)

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 finite-type spaces where $\pi_1$ of the base acts nilpotently on the homology of the fiber: rationalization preserves homotopy fibers.

(This is the fibration lemma orginally due to Bousfield-Kan 72, Chapter II.)

Related concepts

• parametrized homotopy theory

• rational parametrized stable homotopy theory

References

• Aldridge Bousfield, Daniel Kan, Chapter II “Fiber Lemmas” of: Homotopy Limits, Completions and Localizations, Springer 1972 (doi:10.1007/978-3-540-38117-4)

• Steve Halperin, Section 16-20 of: Lectures on minimal models, Mem. Soc. Math. Franc. no 9/10 (1983) (doi:10.24033/msmf.294)

• Flavio da Silveira, Rational homotopy theory of fibrations, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) (pdf)

• 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)

• Yves Félix, Steve Halperin, Jean-Claude Thomas, Sections 4 and 5 of: Rational Homotopy Theory II, World Scientific 2015 (doi:10.1142/9473)