Summary: GENERALIZED PROPERTY R AND THE SCHOENFLIES
ABSTRACT. There is a relation between the generalized Property R Con-
jecture and the Schoenflies Conjecture that suggests a new line of attack
on the latter. The new approach gives a quick proof of the genus 2
Schoenflies Conjecture and suffices to prove the genus 3 case, even in
the absence of new progress on the generalized Property R Conjecture.
1. INTRODUCTION AND PRELIMINARIES
The Schoenflies Conjecture asks whether every PL (or, equivalently, smooth)
3-sphere in S4 divides the 4-sphere into two PL balls. The appeal of the con-
jecture is at least 3-fold:
· The topological version (for locally flat embeddings) is known to
be true in every dimension. Both the PL and the smooth versions
(when properly phrased, to avoid problems with exotic structures)
are known to be true in every other dimension.
· If the Schoenflies Conjecture is false, then there is no hope for a PL
prime decomposition theorem for 4-manifolds, for it would imply
that there are 4-manifolds X and Y, not themselves 4-spheres, so
that X#Y = S4.