Vol. 34, No. 1, 1970

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On the number of nonpiercing points in certain crumpled cubes

Robert Jay Daverman

Vol. 34 (1970), No. 1, 33–43

Let K denote the closure of the interior of a 2-sphere S topologically embedded in Euclidean 3-space E8. If K S is an open 3-cell, McMillan has proved that K has at most one nonpiercing point. In this paper we use a more general condition restricting the complications of K S to describe the number of nonpiercing points. The condition is this: for some fixed integer nK S is the monotone union of cubes with n holes. Under this hypothesis we find that K has at most n nonpiercin g points (Theorem 5). In addition, the complications of K S are induced just by these nonpiercing points. Generally, at least two such points are required, for otherwise n = 0 (Theorem 3).

Mathematical Subject Classification
Primary: 54.78
Received: 4 June 1969
Revised: 14 November 1969
Published: 1 July 1970
Robert Jay Daverman