 
Summary: TopologyVol.23, No. 2, pp. 211217,1984 00409383/84 $3.00+.00
Printedin Great Britain. PergamonPressLtd.
THE FOURDIMENSIONAL SCHOENFLIES CONJECTURE
IS TRUE FOR GENUS TWO IMBEDDINGS
MARTIN SCHARLEMANN~
(Received 1 May 1982)
IT WAS established by Brown[2] that any locallyfiat imbedding of S "~ in S" divides S"
into two domains, each of whose closures is an nball. Somewhat later[5] the heobordism
theorem further established that if S" 1is a smooth of PL submanifold of S" then so are
the resulting nballs, provided that n > 5. (The case n < 3 had been known since
Alexander[l].)
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 3manifold 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 3k to aW~_ t along
~Dk x D 3k, and W~ is obtained from W~by attaching a collar to dW~.
It will be convenient to regard S 4 as the twopoint 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
