This paper proposes to generalize the histories included in Euclidean
functional integrals from manifolds to a more general set of compact
topological spaces. This new set of spaces, called conifolds, includes
nonmanifold stationary points that arise naturally in a semiclasssical
evaluation of such integrals; 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. Postscript figures
available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file
gen1.ps.Comment: 81pp., plain TeX, To appear in Nucl. Phys.