Geometric And Functional Analysis 1016-443X/96/0200268-3351.50+0.20/0 Vol. 6, No. 2 (1996) ~) 1996 Birkh~i.user Verlag, Basel INVARIANTS OF THREE-DIMENSIONAL MANIFOLDS S. GELFAND AND D. KAZHDAN O. Introduction There are several approaches to the construction of invariants of a three- dimensional manifold using quantum groups, and, more generally, monoidal categories. In this paper we consider the combinatorial approach which was suggested by Turaev and Viro (see [TV] and later generalizations in [T2], [KS], [Po], [Ro] among others) and construct an invariant starting from a monoidal category C without any braiding conditions. The starting point of a combinatorial approach is a triangulation of a given three-dimensionai manifold M, and the invariant is constructed using the combinatorics of this triangulation. More precisely, fix a monoidal cat- egory C (a category with the multiplication fimctor X ® Y on objects) over a field k, which is semisimple, has a finite number of isomorphism classes of simple objects, and possesses a duality X ~-~ X* such that X** is iso- morphic to X. To define Ic(M), we must fix a balancing, i.e. functorial isomorphisms X ~ X**, X E Ob C, satisfying certain natural conditions. On the other hand, we do not assume that C satisfies any kind of braiding conditions, i.e. commutativity conditions of the form X®Y ~ Y®X. A sim- ilar construction, which also does not use braiding, was recently suggested by Kuperberg [Ku], who used the la~lguage of Hopf algebras. For any three-dimensional manifold M with boundary S and a trian- gulation D of M, we construct a finite-dimensional linear space W(S, D~) depending on the restriction D ~ of D to S (a triangulation of S) and a vector It(M, S, D) C W(S, D'). The main result of the paper (Theorem 1) is that Ic(M,S,D) depends only on D ~ and not on D itself. Taking the inductive limit of spaces W(S, D ~) over all triangulations D ~ of S, we con- struct a space K(S) and a vector Ic(M, S) C K(S), which is our invariant of a three-dimensionM manifold M with boundary S. (See §6 of the paper, where this is explained in slightly different terms.) In particular, when M is a closed manifold, S = O, we have K(O) = k, and our construction gives an invariant Ic(M) e k. The work of the second author was partially supported by an NSF grant.