2001/09/20 by Frank Quinn, Quinn, Frank
Mathematics · #57R65 57M25 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M25 #msc:57R65
paper · pdf · doi:10.48550/arxiv.math/0109148
21 pages, 4 figures. Revised March 2004 to correct misprints and respond to referee suggestions. Still no worked-out examples, though
arxiv created 2004/03/06 · arxiv updated 2009/11/30
This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index ≤2. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction theory'' using lower commutator quotients. By thinking of a 2-handlebody as essentially determined by the links used as attaching maps for its 2-handles this theory can be thought of a giving ``ambient'' link invariants. The moves used are related to the grope cobordism of links developed by Conant-Teichner, and the Cochran-Orr-Teichner filtration of the link concordance groups. The invariants give algebraically sophisticated ``finite type'' invariants in the sense of Vassilaev.