nLab rational fiber lemma

Idea

The fibration lemma (Bousfield-Kan 72, Chapter II) in (parametrized) rational homotopy theory states sufficient conditions under which rationalization preserves homotopy fibers.

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, for $k$ a ground field of characteristic zero:

1. the fundamental group $\pi_1(B)$ acts nilpotently on the k-rational 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 $A$, $F$ is rationally of finite type,

then the cofiber of any relative Sullivan model for $p$ is a Sullivan model for $F$ (both as dgc-algebras over $k$).

(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)

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.)

Related concepts

• parametrized homotopy theory

• rational 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)

• 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)