The Four - Dimensional

IT WAS established by Brown[2] that any locally-fiat imbedding of S "-~ in S" divides S" into two domains, each of whose closures is an n-ball. Somewhat later[5] the h-eobordism theorem further established that if S"1 is a smooth of P L submanifold of S" then so are the resulting n-balls, 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 3-manifold M will be a decomposition = W~ c W1 c W~ c W2 ~ W~ ,-" . . . c W , _ 1 c 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 ~D k 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 b e the standard projections. A P L imbedding g: $3-,S 3 x R ~ S 4 is a critical level imbedding if there is a collared handlebody decomposition /4/"1 = W~ c . . . c W~ of g(S 3) and a collection tj < . . . < t~ for plg(S 3) such that, for l < _ i < n , h ~ f f i ( S 3 x { t ~ } ) f 3 g ( S 3) and, for t ~ < t < t ~ + l , n((S 3 x { t } ) f q g ( S 3 ) ) = n(g(~W~)). Thus the handles o f g ( S 3) are all horizontal in S 3 x R and the collars are all vertical. It is known that any P L imbedding is P L isotopic to a critical level imbedding[3]. A critical level imbedding g: $ 3 ~ S 4 is said to have genus n if the associated handlebody decomposition of S 3 has k 0-handles and (k + n 1) 1-handles. The goal here is to show that any P L critical level imbedding g: $3---,S 4 of genus < 2 is standard, that is, it is P L ambient isotopic to the standard imbedding. We work in the P L category exclusively; all maps, imbeddings and isotopies are understood to be PL.