Appropriate cross-sectionally simple four-cells are flat

Steven Alan Pax

Vol. 108 (1983), No. 2, 379–384

DOI: 10.2140/pjm.1983.108.379

Abstract

When X is a set in Eⁿ, we let X_t = X ∩H_t—where H_t is the horizontal hyperplane in Eⁿ of height t. In this note, we prove that a 4-cell B in E⁴, such that each nonempty slice B_t is either a point or a 3-cell, is flat whenever, for all t, B_t is flat in H_t and Bd B_t is flat in Bd B.

Mathematical Subject Classification 2000

Primary: 57N13

Secondary: 57N45, 57N50

Milestones

Received: 19 August 1981

Revised: 14 January 1983

Published: 1 October 1983