On doubled 3-manifolds and minimal handle presentations for 4-manifolds

We extend our earlier work on free reduction problems for 2-complexes K in 4-manifolds N (i.e., the problem of effecting, by a geometric deformation of K in N , the free reduction of the relator words in the presentation associated with K). Here, the problem is recast, with new results, in terms of 2-handle presentations of 4-manifolds. Let M∗ be the complement of the interior of a closed 3-ball in the 3-manifold M , and let 2M∗ be the connected sum of two copies M , via a boundary identification allowing the identification of 2M∗ with the boundary of M∗ × [−1, 1]. We show that algebraic handle cancellation associated with a 2handle presentation of a 4-manifold with boundary 2M∗ can be turned into geometric handle cancellation for handle presentations of possibly different 4-manifolds having the same boundary provided that certain obstruction conditions are satisfied. These conditions are identified as surgery equivalence classes of framed links in Bd(M∗ × [−1, 1]). These links, without the framing information, were considered in previous work by the author. The following is one of the main results here: Let M be a 3-manifold that is a rational homology sphere, and suppose that M∗× [−1, 1] has a handle presentation H with no handles of index greater than 2. Suppose H is a normal, algebraically minimal handle presentation. If the obstruction conditions are satisfied, then there is a 4-manifold N bounded by 2M∗ that has a minimal handle presentation. Another theorem states, independent of the Poincaré Conjecture, conditions for a homotopy 3-sphere to be S in terms of minimal handle presentations and the triviality of the defined obstruction conditions.