Homology surgery and invariants of 3–manifolds

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 $\bar{μ}$ –invariants to this class of links.

Keywords

homology surgery, finite type invariants, 3–manifolds, clovers

Mathematical Subject Classification 2000

Primary: 57N10

Secondary: 57M25

References

Forward citations

Publication

Received: 31 May 2000

Revised: 2 May 2001

Published: 17 June 2001

Proposed: Robion Kirby

Seconded: Joan Birman, Cameron Gordon

Authors

Stavros Garoufalidis
School of Mathematics Georgia Institute of Technology Atlanta Georgia 30332-0160 USA

Jerome Levine
Department of Mathematics Brandeis University Waltham Massachusetts 02254-9110 USA

Volume 5, issue 2 (2001)

Stavros Garoufalidis and Jerome Levine