Abstract Quantum amplitudes for euclidean gravity constructed by sums over compact manifold histories are a natural arena for the study of topological effects. Such euclidean functional integrals in four dimensions include histories for all boundary topologies. However, a semiclassical evaluation of the integral will yield a semiclassical amplitude for only a small set of these boundaries. Moreover, there are sequences of manifold histories in the space of histories that approach a stationary point of the Einstein action but do not yield a semiclassical amplitude; this occurs because the stationary point is not a compact Einstein manifold. Thus the restriction to manifold histories in the euclidean functional integral eliminates semiclassical amplitudes for certain boundaries even though there is a stationary point for that boundary. In order to incorporate the contributions from such semiclassical histories, this paper proposes to generalize the histories included in euclidean functional integrals a more general set of compact topological spaces. This new set of spaces, called conifolds, includes the nonmanifold stationary points; additionally, it can be proven that sequences of approximately Einstein manifolds and sequences of approximately Einstein conifolds both converge to Einstein conifolds. Consequently, generalized euclidean functional integrals based on these conifold histories yield semiclassical amplitudes for sequences of both manifold and conifold histories that approach a stationary point of the Einstein action. Therefore sums over conifold histories provide a useful and self-consistent starting point for further study of topological effects in quantum gravity.