Fixed Points of Fiber Maps of Montgomery-samelson Fiberings over Intervals
نویسنده
چکیده
A Montgomery-Samelson (MS) fibering is a fibering with singularities such that the singular fibers are points. A fiber map is a map that preserves fibers and takes singular fibers to singular fibers. For MS fiberings that are the suspension of a map of a space to a point, and hence the base space is an interval, we obtain a formula for the minimum number of fixed points among all fiber maps that are fiber homotopic to a fiber map from the fibering to itself. The formula calculates that minimum number in terms of the minimum number of fixed points among the maps homotopic to self-maps of nonsingular fibers that are restrictions of the fiber map. These minimum numbers can often be computed by Nielsen fixed point theory. We present an example of an MS fibering in which the total space and nonsingular fibers are simply-connected and yet, in contrast to fiber maps of fiberings without singularities, for each natural number k there is a fiber map f [k] such that the minimum number of fixed points of the fiber maps fiber homotopic to f [k] is k. Subject Classification 55M20, 55R55
منابع مشابه
On Compact Fiberings of the Plane
In a recent paper [2], Montgomery and Samelson have raised the question whether there exists, for some n, a compact fibering of Euclidean w-space, and have given some reasons for thinking that no such fibering is possible. The purpose of this note is to provide further evidence for this belief by proving that a t least there is no compact fibering of the plane. The theorem I shall prove is in f...
متن کاملOne-parameter Transformation Groups of the Three-sphere
Introduction. In [3], Seifert describes all the fiberings of the three-sphere. The fibers are the orbits when the circle group acts effectively on the three-sphere without fixed points in a certain way. In [2,§ 6.7], Montgomery and Zippin show that when the circle group acts on the three-sphere effectively but with fixed points its action is described in a suitable coordinate system as a rotati...
متن کاملA RESULT ON FIXED POINTS FOR WEAKLY QUASI-CONTRACTION MAPS IN METRIC SPACES
In this paper, we give a new fixed point theorem forWeakly quasi-contraction maps in metric spaces. Our results extend and improve some fixed point and theorems in literature.
متن کاملSurface Bundles over Surfaces with Arbitrarily Many Fiberings
In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each n ≥ 3 we construct 4-manifolds E admitting at least n distinct fiberings pi : E → Σgi as a surface bundle over a surface with base and fiber both closed surfaces of negative Euler characteristic. We give examples of surface bundles admitting multiple fiberings for which th...
متن کامل