On the combinatorial Schoenflies conjecture

Abstract

Introduction. The combinatorial version of the Schoenflies conjecture in dimension n states: A combinatorial (n 1)-sphere on a combinatorial n-sphere decomposes the latter into two combinatorial ncells. The cases n =1, 2 are obvious, the case n =3 was solved by J. W. Alexander [1], see also W. Graeub [5], and E. Moise [8]. Nothing is known for n > 3 (compare J. F. Hudson and E. C. Zeeman in [6, p. 729]). On the other hand the following form of the Hauptvermutung for spheres is proved: If a combinatorial n-dimensional manifold is homeomorphic an n-dimensional sphere, then it is a combinatorial n-sphere if n04. This was proved by S. Smale for n $4, 5, 7 in [9], and improved by E. C. Zeeman to n #4 (unpublished). I would like to thank Professor Zeeman for pointing out this result to me. Further a generalized Schoenflies theorem was proved by M. Brown in [3]. If this Hauptvermutung were true for all n, a simple induction on the dimension n would prove the combinatorial Schoenflies conjecture for all n (compare Theorem 6). In the following we show that either the combinatorial Schoenflies conjecture is true for all dimensions n, or it is false for all dimensions n> 3.