Local to global theorems in the theory of Hurewicz fibrations

Abstract

Abstract. This paper is concerned with the problem of showing a local fibration is a fibration. There are two kinds of local to global theorems proven. The first type of theorem considers local fibrations where local is in terms of closed covers of the base (e.g. the set of closed simplices of a polyhedron, the cones of a suspension). The second type of theorem deals with local in terms of open covers of the total space.1. Introduction. The now classical Uniformization Theorem in the theory of fibrations [1] states that in the paracompact situation a local fibration is a fibration, where local is in terms of an open cover of the base. One of the main objectives of this paper is to derive similar local to global theorems in cases where the covers are closed, e.g. the set of closed simplices of a polyhedron. In addition, we also prove (with more generous hypothesis) the Hurewicz fibrations version of a result of Cheeger and Kister [2], that a local local disk bundle is a bundle.

Attaching Hurewicz fibrations with fiber preserving maps

“…This can be done exactly as in the Lifting Extension Theorem in [1], replacing λ^ and Ω PA by λ 0 and Ω Po throughout the proof, and noting that (E, E o ) a closed cofibration implies (Ω p , Ω p ) is a closed cofibration. In fact if we let π λ : Ω p -> E be projection on the first coordinate, then Γ = (Ω p , π l9 E) is a fibration (Γ = the pullback by p of the standard path fibration over B), and since Ω PQ -πϊ^Eo), (Ω p , Ω Po ) is a closed cofibration by Theorem 12 of [8].…”

“…If ξ has an extension I = (E 9 p, B), consider the space over B x {0} U A x I Since (A x /, A x {0}) and (5 x {0}, A x {0}) are closed cofibrations, Theorem (4.2) of [1] applies, and ξ \J Γ is a fibration Let r: B x ί^5x(0}uAx I he a retraction. Then r*(f U Γ), the pullback of f U Γ by r, is a fibration, and the desired extension of η is given by r*(ξ U Γ) BX{1) .…”

“…We now apply Theorem (3.6) and Theorem (4.2) of [1] to prove a stronger form of Dold's pasting lemma. (See §6.4,of [2].)…”

“…If in addition there is a lifting function λ: Ω p -* E 1 for f such that X\Ω PQ is a lifting function for ξ 0 , then (ζ, ξ Q ) will be called a fibered pair.…”