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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Dual decompositions of 4-manifolds II: Linear invariants

Author: Frank Quinn
Journal: Trans. Amer. Math. Soc. 358 (2006), 2161-2181
MSC (2000): Primary 57R65, 57M25
Published electronically: May 26, 2005
MathSciNet review: 2197452
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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.

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Frank Quinn
Affiliation: Department of Mathematics, Virginia Polytechnical Institute and State University, Blacksburg, Virgina 24061-0123

Received by editor(s): December 10, 2001
Received by editor(s) in revised form: May 11, 2004
Published electronically: May 26, 2005
Additional Notes: This work was partially supported by the National Science Foundation
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.