Proving that wild cells exist

P. H. Doyle, III and John Gilbert Hocking

Vol. 27 (1968), No. 2, 265–266

DOI: 10.2140/pjm.1968.27.265

Abstract

In their famous paper Fox and Artin constructed several examples of wild cells in 3-space. The present authors construct a wild disk D in the 4-sphere S⁴ with the property that the proof of nontameness is perhaps the most elementary possible. We require only the knowledge that if K is the trefoil knot in the 3-sphere S³, then the fundamental group π₁(S³ −K) is not abelian. Parenthetically, the wild disk D constructed here has the property that every arc on D is tame, a fact which follows immediately from the construction.

Mathematical Subject Classification

Primary: 54.78

Milestones

Received: 4 January 1968

Published: 1 November 1968