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Homology surgery and invariants of 3–manifolds

Stavros Garoufalidis and Jerome Levine

Geometry & Topology 5 (2001) 551–578

arXiv: math.GT/0005280


We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of π–algebraically-split links in 3–manifolds with fundamental group π. Using this class of links, we define a theory of finite type invariants of 3–manifolds in such a way that invariants of degree 0 are precisely those of conventional algebraic topology and surgery theory. When finite type invariants are reformulated in terms of clovers, we deduce upper bounds for the number of invariants in terms of π–decorated trivalent graphs. We also consider an associated notion of surgery equivalence of π–algebraically split links and prove a classification theorem using a generalization of Milnor’s μ̄–invariants to this class of links.

homology surgery, finite type invariants, 3–manifolds, clovers
Mathematical Subject Classification 2000
Primary: 57N10
Secondary: 57M25
Forward citations
Received: 31 May 2000
Revised: 2 May 2001
Published: 17 June 2001
Proposed: Robion Kirby
Seconded: Joan Birman, Cameron Gordon
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
Georgia 30332-0160
Jerome Levine
Department of Mathematics
Brandeis University
Massachusetts 02254-9110