  We investigate the influence of the chimney topology $T\times T\times R$ of
the Universe on the gravitational potential and force that are generated by
point-like massive bodies. We obtain three distinct expressions for the
solutions. One follows from Fourier expansion of delta functions into series
using periodicity in two toroidal dimensions. The second one is the summation
of solutions of the Helmholtz equation, for a source mass and its infinitely
many images, which are in the form of Yukawa potentials. The third alternative
solution for the potential is formulated via the Ewald sums method applied to
Yukawa-type potentials. We show that, for the present Universe, the formulas
involving plain summation of Yukawa potentials are preferable for computational
purposes, as they require a smaller number of terms in the series to reach
adequate precision.
