Stavros Garoufalidis and J. Levine.
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 $\bar\mu$-invariants to this class of links.
Key words: Homology surgery, finite type invariants, 3-manifolds, clovers.
Notes: 14 pages with 15 figures.