New York Journal of Mathematics
Volume 18 (2012) 29-53


R. Craggs

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

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Published: January 31, 2012
Keywords: framed surgery, extended Nielsen operations, handle presentations, graph manifolds
Subject: Primary: 57M20; Secondary: 57R65, 57M40

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 2-handle 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 S3 in terms of minimal handle presentations and the triviality of the defined obstruction conditions.

Author information

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green, Urbana, IL 61801