On Finite Type 3-manifold Invariants I

Recently Ohtsuki Oh2], motivated by the notion of nite type knot invariants, introduced the notion of nite type invariants for oriented, integral ho-mology 3-spheres (ZHS for short). In the present paper we propose another definition of nite type invariants of Z HS and give equivalent reformulations of our notion. We show that our invariants form a ltered commutative algebra and are of nite type in in the sense of Ohtsuki and thus conclude that the associated graded algebra is a priori nite dimensional in each degree. We discover a new set of restrictions that Ohtsuki's invariants satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the nite type 3-manifold invariants with the (Vassiliev) knot invariants.