Examples of singular normal complex spaces which are topological manifolds.

In case V is irreducible with singular curve C, we cannot write every oeH3(W V) as a tube. Indeed, oa = 6C4 and C4 Will meet C in a finite number of points. The trouble is similar to the above and may be overcome as follows: Let DC V be a curve meeting C transversely at a finite number of points pi,...pt. For simplicity, assume t = 1, p = pi, and let B be a small ball around p in W. We may assume that, locally, D is Vie (2-plane in 3-space) and we restrict our attention to V(this 2-plane). We may construct a tube r, over D -D n B,, and 6rT, will be a finite number (= number of local branches of V) of linked toral surfaces in a 3sphere. If we take out the solid tori from the sphere, we are left with a 3-chain a' such that T, + fT = T is a 3-cycle in W V. Furthermore, T = 6c4 where c4C = {p}. Then, if aoH3(W V), a kr will be a 3-cycle with ukr = bc4 and c4. C = 0. Thus, a kT is a linear combination of tubes and, to evaluate f, co, we may evaluated c. This integral may, as before, be written as a Cauchy integral.