  We perform a Hamiltonian reduction of spherically symmetric Einstein gravity
with a thin dust shell of positive rest mass. Three spatial topologies are
considered: Euclidean (R^3), Kruskal (S^2 x R), and the spatial topology of a
diametrically identified Kruskal (RP^3 - {a point at infinity}). For the
Kruskal and RP^3 topologies the reduced phase space is four-dimensional, with
one canonical pair associated with the shell and the other with the geometry;
the latter pair disappears if one prescribes the value of the Schwarzschild
mass at an asymptopia or at a throat. For the Euclidean topology the reduced
phase space is necessarily two-dimensional, with only the canonical pair
associated with the shell surviving. A time-reparametrization on a
two-dimensional phase space is introduced and used to bring the shell
Hamiltonians to a simpler (and known) form associated with the proper time of
the shell. An alternative reparametrization yields a square-root Hamiltonian
that generalizes the Hamiltonian of a test shell in Minkowski space with
respect to Minkowski time. Quantization is briefly discussed. The discrete mass
spectrum that characterizes natural minisuperspace quantizations of vacuum
wormholes and RP^3-geons appears to persist as the geometrical part of the mass
spectrum when the additional matter degree of freedom is added.
