nLab rational fiber lemma




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

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

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

If, for kk a ground field of characteristic zero:

  1. the fundamental group π 1(B)\pi_1(B) acts nilpotently on the k-rational homology groups H (F,k)H_\bullet(F,k)

    (e.g. if BB is simply connected, or if the fibration is a principal bundle),

  2. at least one of AA, FF is rationally of finite type,

then the cofiber of any relative Sullivan model for pp is a Sullivan model for FF (both as dgc-algebras over kk).

(Félix-Halperin-Thomas 00, Theorem 15.3, following Halperin 83, Section 16)

Moreover, if CE(𝔩B)CE(\mathfrak{l}B) is a minimal Sullivan model for BB, then the cofiber of the corresponding minimal relative Sullivan model for pp is the minimal Sullivan model CE(𝔩F)CE(\mathfrak{l}F) for FF:

(1)CE(𝔩F) cofib(CE(𝔩p)) CE(𝔩 BA) CE(𝔩p) CE(𝔩B) \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.)


Last revised on July 16, 2021 at 10:19:09. See the history of this page for a list of all contributions to it.