nLab
Hurewicz fibration

Serre fibration \Leftarrow Hurewicz fibration \Rightarrow Dold fibration \Leftarrow shrinkable map


Contents

Definition

Definition

A continuous map p:EBp \;\colon\; E\longrightarrow B of topological space is called a Hurewicz fibration it it satisfies the right lifting property with respect to maps of the form σ 0:XX×{0}X×I\sigma_0 \;\colon\; X\cong X\times\{0\}\hookrightarrow X\times I for all topological spaces XX.

Remark

This right lifting property is in this context called the homotopy lifting property, because the maps from X×IX\times I are understood as homotopies. In more detail, for every space XX, any homotopy F:X×IBF:X\times I\to B, and a continuous map f:XEf:X\to E, there is a homotopy F˜:X×IE\tilde{F}:X\times I\to E such that f=F˜σ 0:=F˜ 0f =\tilde{F} \circ\sigma_0 :=\tilde{F}_0 and F=pF˜F=p\circ\tilde{F}:

X f E σ 0 F˜ p X×I F B. \array{ X &\stackrel{f}\to& E \\ \downarrow^{\sigma_0} &{}^{\tilde{F}}\nearrow& \downarrow^p \\ X\times I &\stackrel{F}{\to}& B } \,.
Remark

Strictly speaking, the “all” in this context should be interpreted to refer to all spaces in whatever ambient category of spaces one is working in, since frequently this is a convenient category of spaces. In theory, therefore, a map in such a category could be a Hurewicz fibration in that category without necessarily being a Hurewicz fibration in the category of all topological spaces, but in practice this usually doesn’t make a whole lot of difference.

Instead of checking the homotopy lifting property, one can instead solve a universal problem:

Theorem

A map is a Hurewicz fibration precisely if it admits a Hurewicz connection. (See there for details.)

Appearance in a model structure

There is a Quillen model category structure on Top where fibrations are Hurewicz fibrations, cofibrations are closed Hurewicz cofibrations and weak equivalences are homotopy equivalences; see model structure on topological spaces and Strøm's model category. There is a version of Hurewicz fibrations for pointed spaces, as well as in the slice category Top/B 0Top/B_0 where B 0B_0 is a fixed base.

References

The historical paper of Hurewicz is

  • Witold Hurewicz, On the concept of fiber space, Proc. Nat. Acad. Sci. USA 41 (1955) 956–961; MR0073987 (17,519e) PNAS,pdf.

A decent review of Hurewicz fibrations, Hurewicz connections and related issues isin

  • James Eells, Jr., Fibring spaces of maps, in Richard Anderson (ed.) Symposium on infinite-dimensional topology

A textbook account of the homotopy lifting property is for instance in

See also the textbooks on algebraic topology by Whitehead and Spanier.

Revised on December 8, 2013 02:10:07 by Urs Schreiber (89.204.135.58)