Embedding spheres and balls in codimension $\leqq 2$

Abstract

regular neighborhood of (jV=lS~-l ) v (k~lST,-2 ) with codimension one as an addendum to Theorem 2.1. Addendum to Theorem 2.1. Let N be a compact q-manifold. Suppose that q>=6. 7hen N is an abstract regular neighborhood of jV v kVISZ, i f N is ( q 3)-connected and ON consists o f r + l connected components Mo, M1 . . . . . M, whose fundamental groups rq(Ml) are of weights r~ equal to the Betti numbers of HI(M.t ). (For the weight of a group, see [18].) Remark. One more effort shows that the converse of the above is true for q_~ 4. Furthermore, if the 4-dimensional Poincar6 conjecture is Embedding Spheres and Balls in Codimension ~ 2 97 true, then the above is true for q= 5. Notice that the images of S~ -~ and S~-2 represent a basis of/2/. (N). This fact was the starting point of our investigation of the paper. (Refer to [16], Theorem 3.12.) Proof. For each j, let ;tjk, k= 1, ..., rj, be elements of ~l(Mi) whose normal closure equals ~I(Mj). Since r~ equals the Betti number of HI(Mj), it follows that the Hurewicz images of ,;tjk form a basis of HI(M~). Thus in the proof of the structure theorem we may take f2 k SO that f2 k] O q2 X S ~ represent ~-jk. Therefore, in the handle decomposition of N, handles (ffl-2) kill the fundamental groups ~(Mj) and hence C can be taken so that OC is simply connected. Since q>6, it follows from Poincar6Smale theorem that C is a q-ball. Consequently, N is obtained from the q-ball D ~= C by attaching to S~-~cD ~ handles of indices q 1 and q 2 via a framed link f : r S q 2 x D w t S ~3x D2--* S q-l. Now a subpolyhedron 0 * f (r S q2 x O w t S q3 x 02) w ((r D ~-1 x O) w (t D q2 x 02)) of N is the required spine of N, where 0 9 X denotes the cone of X S q~ from the origin 0, completing the proof. w 3. Nuli-Cobordism of Links in Homology Spheres Let C be a contractible (m+ 1)-manifold. Definition of Null-Cobordism. A link f : S (n, rn) ~ ~C is null-cobordant, if it can be extended to a locally fiat embedding F: D(n + 1, r , ) ~ O C, called a null-cobordism off. A framed link f : ~ S ( n , rn)--.aC is nullcobordant, if it can be extended to an embedding F: ~m+ 1D(n + 1, r,)--* C, called a null-cobordism of f, such that F(~ '~+ID(n+ 1, r,))c~ t~C= F ( ~ S (n, rn) ). Notice that in case m => 4, since by Lemma 2.2 a homology m-sphere bounds a unique contractible (m+ 1)-manifold, for a (framed) link the notion of null-cobordism makes sense without mentioning a contractible manifold bounding the homology sphere. It follows again from Lemma 2.2, (1) that if f is null-cobordant, then a link equivalent t o f i s also nullcobordant, provided at least m~ 3 or C= D m+~. We shall be concerned mainly with the case m n < 2. In that case, by the product neighborhood theorem of locally flat spheres in a homology sphere with codimension < 2 (for example refer to [31]), a link f : r S ~l w t S m2 ~ M is always framed and two framed links are equivalent if and only if their underlying links are equivalent, provided m 2 ~ : 1. Thus a framed link of codimension < 2 is null-cobordant if and only if its underlying link is null-cobordant. Moreover, the cone extension argument of a homeomorphism of S n shows that a link is null-cobordant if the image spheres in 0C bound mutually disjoint locally fiat balls in C.