  We study the properties of the Newtonian gravitational potential in a
spherical Universe for different topologies. For this, we use the non-Euclidean
Newtonian theory developed in Vigneron [2022, Class. & Quantum Gravity, 39,
155006] describing Newtonian gravitation in a spherical or hyperbolic Universe.
The potential is calculated for a point mass in all the globally homogeneous
regular spherical topologies, i.e. whose fundamental domain is unique and is a
platonic solid. We provide the exact solution and the Taylor expansion series
of the potential at a test position near the point mass. We show that the odd
terms of the expansion can be interpreted as coming from the presence of a
non-zero spatial scalar curvature, while the even terms relate to the closed
nature of the topological space. A consequence is that, compared to the point
mass solution in a 3-torus, widely used in Newtonian cosmological simulations,
the spherical cases all feature an additional attractive first order term
dependent solely on the spatial curvature. The choice of topology only affects
the potential at second order and higher. For typical estimates of cosmological
scales (curvature and topology), the strongest topological effect occurs in the
case of the Poincar\'e dodecahedral space, but in general the effect of
curvature dominates over topology. We also provide the set of equations that
can be used to perform $N$-body simulations of structure formation in spherical
topologies.
