 

R. Craggs
On doubled 3manifolds and minimal handle presentations for 4manifolds view print


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


Abstract
We extend our earlier work on free reduction problems for 2complexes K in 4manifolds 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 2handle presentations of 4manifolds.
Let M_{∗} be the complement of the interior of a closed 3ball in the 3manifold 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 4manifold with boundary 2M_{∗} can be
turned into geometric handle cancellation for handle presentations of possibly different 4manifolds 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 3manifold 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 4manifold N bounded by 2M_{∗} that has a minimal
handle presentation.
Another theorem states, independent of the Poincaré Conjecture, conditions for a homotopy 3sphere to be S^{3} in terms of minimal handle presentations and the triviality of the defined obstruction conditions.


Author information
Department of Mathematics, University of Illinois at UrbanaChampaign, 1409 W. Green, Urbana, IL 61801
craggs@math.uiuc.edu

