GENERALIZED SUMS OVER HISTORIES FOR QUANTUM-GRAVITY .2. SIMPLICIAL CONIFOLDS

This paper examines the issues involved with concretely implementing a sum over conifolds in the formulation of euclidean sums over historie s for gravity. The first step in precisely formulating any sum over to pological spaces is that one must have an algorithmically implementabl e method of generating a list of all spaces in the set to be summed ov er. This requirement causes well known problems in the formulation of sums over manifolds in four or more dimensions; there is no algorithmi c method of determining whether or not a topological space is an n-man ifold in five or more dimensions and the issue of whether or not such an algorithm exists is open in four. However, as this paper shows, con ifolds are algorithmically decidable in four dimensions. Thus the set of 4-conifolds provides a starting point for a concrete implementation of euclidean sums over histories in four dimensions. Explicit algorit hms for summing over various sets of 4-conifolds are presented in the context of Regge calculus.