On the Concept of Fiber Space

Abstract

MATHEMATICS: W. HUREWICZ years (private communication), that the solutions given in Table 2 are the only possible solutions in positive integer values of n and w (n > w) of equation (8). Dr. W. Ljunggren (private communication) has independently confirmed Dr. Skolem's statement. The three lower pairs of values given in Table 2 satisfy equation (8) and are given in parentheses for completeness, but are not in agreement with Gaunt's condition n > 2. When n > 2, Table 2 represents supplementary selection rules to those found by Gaunt for integral (7). I am indebted to Drs. Th. Skolem and W. Ljunggren, of the Institutes of Mathematics of the Universities of Oslo and Bergen, respectively, for solving the numbertheory problems in this discussion and to Mr. Oddmund Kolberg for suggesting the transformation of equation (8) into equation (9) that made it possible to use standard methods for solving equation (8).

The Lusternik–Schnirelmann category and the fundamental group

“…U is a trivial stratified bundle and hence k is a Hurewicz fibration over f 1 .U / for all U 2 U . Then by Hurewicz [9] (see also Dold [4]) we conclude that k is a Hurewicz fibration over X .…”

The equivalence of fiber spaces and bundles

“…Introduction. The objective of this paper is to verify the conjecture made in [2] that every Hurewicz fibration [3] over a polyhedral base is fiber homotopy equivalent to a Steenrod fiber bundle [6]. The result relies heavily on Milnor's universal bundle construction [4] and the following extension [2] of a theorem of A. Dold [l].…”

Local to global theorems in the theory of Hurewicz fibrations

“…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.…”

“…Furthermore, since X is metric, we can assume Xx and A2 are regular. By the Uniformization Theorem [1], we need only show that every point in A'has a neighborhood C/such that £u is a Hurewicz fibration, and obviously it is sufficient to consider points in Xx n X2 every neighborhood of which intersects X-x -Xx n X2 and X2 -Xx n X2 nontrivially.…”

“…The argument used to prove Theorem 4.2 will produce a regular lifting function a2 with the additional property that (1) ViPaÁe, a>)(t) = 7Txp(e).…”