 
Summary: GENERALIZED PROPERTY R AND THE SCHOENFLIES
CONJECTURE
MARTIN SCHARLEMANN
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)
3sphere in S4 divides the 4sphere into two PL balls. The appeal of the con
jecture is at least 3fold:
· 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 4manifolds, for it would imply
that there are 4manifolds X and Y, not themselves 4spheres, so
that X#Y = S4.
