Decompositions of E³ which yield E³

Ralph Joseph Bean

Vol. 20 (1967), No. 3, 411–413

DOI: 10.2140/pjm.1967.20.411

Abstract

In recent years interest has been focused on the following two questions.

If G is an upper semi-continuous decomposition of E³ whose decomposition space G′ is homeomorphic to E³, under what conditions can we conclude that

(1) each element of G is point-like?

(2) there is a pseudo-isotopy F : E³ × [0,1] → E³ such that F|E³ × 0 is the identity and F|E³ × 1 is equivalent to the projection map Π : E³ → G′?

An example of Bing of a decomposition of E³ into points, circles, and figure-eights shows that some additional hypotheses must be inserted. The theorem presented here gives such hypotheses, namely that the nondegenerate elements form the intersection of a decreasing swquence of finite disjoint unions of cells-withhandles, and project into a Cantor set.

Mathematical Subject Classification

Primary: 54.78

Milestones

Received: 9 August 1965

Published: 1 March 1967