Homology surgery and invariants of 3–manifolds

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