  Using Monte Carlo simulations we study the two-dimensional Ising model on
triangular, square, and hexagonal lattices with various topologies. We focus on
the behavior of the magnetic susceptibility and of the specific heat near the
critical point of the planar bulk system. We find that scaling functions of
these quantities on the spherical surface (Euler characteristic K = 2) differ
from the scaling functions on the projective plane (K = 1) which, in turn,
differ from the scaling functions on the torus and on the Klein bottle (both K
= 0). This provides strong evidence that phase transitions of the Ising model
on two-dimensional surfaces depend on their topologies.
