Dual decompositions of 4-manifolds II: Linear invariants
Trans. Amer. Math. Soc. 358 (2006), 2161-2181
Primary 57R65, 57M25
May 26, 2005
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Abstract: This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index . 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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- Tim Cochran and Paul Melvin, Finite type invariants of 3-manifolds, math.GT/9805026, Invent. Math. 140 (2000), 45-100. MR 1779798 (2002a:57015)
- Tim Cochran, Kent Orr, and Peter Teichner, Knot concordance, Whitney towers, and signatures, math.GT/9908117, Ann. of Math. 157 (2003), 433-519. MR 1973052 (2004i:57003)
- James Conant and Peter Teichner, Grope cobordism of classical knots, Topology 43 (2004), 119-156. MR 2030589 (2004k:57006)
- Michael Freedman and Frank Quinn, Topology of 4-manifolds, Princeton University Press, 1990.MR 1201584 (94b:57021)
- Frank Quinn, Dual decompositions of 4-manifolds, Trans. Amer. Math. Soc. 354 (2002), 1373-1392.MR 1873010 (2002k:57080)
- Andrew Ranicki, Noncommutative localization in topology (to appear).
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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
May 26, 2005
This work was partially supported by the National Science Foundation
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.