#### Volume 5, issue 2 (2001)

 Download this article For printing
 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Homology surgery and invariants of 3–manifolds

### Stavros Garoufalidis and Jerome Levine

Geometry & Topology 5 (2001) 551–578
 arXiv: math.GT/0005280
##### Abstract

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 $\pi$–algebraically-split links in 3–manifolds with fundamental group $\pi$. 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 $\pi$–decorated trivalent graphs. We also consider an associated notion of surgery equivalence of $\pi$–algebraically split links and prove a classification theorem using a generalization of Milnor’s $\stackrel{̄}{\mu }$–invariants to this class of links.

##### Keywords
homology surgery, finite type invariants, 3–manifolds, clovers
Primary: 57N10
Secondary: 57M25
##### 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