  The role of topology in elementary quantum physics is discussed in detail. It
is argued that attributes of classical spatial topology emerge from properties
of state vectors with suitably smooth time evolution. Equivalently, they emerge
from considerations on the domain of the quantum Hamiltonian, this domain being
often specified by boundary conditions in elementary quantum physics. Several
examples are presented where classical topology is changed by smoothly altering
the boundary conditions. When the parameters labelling the latter are treated
as quantum variables, quantum states need not give a well-defined classical
topology, instead they can give a quantum superposition of such topologies. An
existing argument of Sorkin based on the spin-statistics connection and
indicating the necessity of topology change in quantum gravity is recalled. It
is suggested therefrom and our results here that Einstein gravity and its minor
variants are effective theories of a deeper description with additional novel
degrees of freedom. Other reasons for suspecting such a microstructure are also
summarized.
