Summary: TopologyVol.23, No. 2, pp. 211-217,1984 0040-9383/84 $3.00+.00
Printedin Great Britain. PergamonPressLtd.
THE FOUR-DIMENSIONAL SCHOENFLIES CONJECTURE
IS TRUE FOR GENUS TWO IMBEDDINGS
(Received 1 May 1982)
IT WAS established by Brown that any locally-fiat imbedding of S "-~ in S" divides S"
into two domains, each of whose closures is an n-ball. Somewhat later the h-eobordism
theorem further established that if S"- 1is a smooth of PL submanifold of S" then so are
the resulting n-balls, provided that n > 5. (The case n < 3 had been known since
For n = 4 little is known. The goal of this paper is to present an elementary proof of
the conjecture for the special ease described below.
A collared handlebody decomposition of a 3-manifold M will be a decomposition
= W~ c W1 c W~ c W2 ~ W~ ,-" ... c W,_ 1c W;_ 1~ Wn ~- M such that, for
0 < i < n, Wi is obtained from W~_~by attaching a handle hi ~- D t × D 3-k to aW~_ t along
~Dk x D 3-k, and W~ is obtained from W~by attaching a collar to dW~.
It will be convenient to regard S 4 as the two-point compactification of S 3 × R, so
S 3 x ~ ~ S 4. Let p: S 3 x ~.-.).R, n: S 3 × ~...-~S 3 be the standard projections. A PL
imbedding g: $3--,S 3 x R ~ S 4 is a critical level imbedding if there is a collared handle-