Vol. 20, No. 3, 1967

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Decompositions of E3 which yield E3

Ralph Joseph Bean

Vol. 20 (1967), No. 3, 411–413

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 Gis 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
Received: 9 August 1965
Published: 1 March 1967
Ralph Joseph Bean