Dual decompositions of 4-manifolds II: Linear invariants

Abstract:

This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq 2$. Part I (Trans. Amer. Math. Soc. 354 (2002), 1373–1392) 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 as 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.