On attaching 3-handles to a 1-connected 4-manifold
Bruce Trace · Pacific Journal of Mathematics · 1982
This paper studies various properties of 3-handle presentations of 1-connected, smooth, 4-manifolds-with-connectedboundary.Although the question of whether or not such 4-manifolds have handle presentations consisting solely of 0-, 1-and 2-handles remains open, the main results of this paper do imply that the essential structure of such 4-manifolds is contained in a neighborhood of their 2-skeleton.Throughout this article all maps and manifolds will be of class C 00 .The key idea exploited here is that a knot in the boundary of a 4-manifold which is slice in this 4-manifold (i.e., which bounds a smooth, property embedded 2-disc) may be viewed as the cocore of a 2-handle -the 2-handle being a tubular neighborhood of the slice disc.Thus, if ikP is obtained from W^ by attaching a 3-handle, h\ along an embedded Swhere Σ 2 is the image of S 2 x {0} under this embedding, then in order to construct a 2-handle which is complementary to h z we need only construct a knot KcdW* which meets Σ 2 transversely in a single point (i.e., K and Σ 2 are complementary in dW 4 ) and which is slice in W\ If the boundary of W* is connected, the existence of a knot KadW* which is complementary to Σ 2 is clearly equivalent to Σ 2 not separating dW*.If W* is 1-connected, the following proposition tells us that a knot complementary to Σ 2 must be slice.