  The behaviour of the mean Euler-Poincar\'{e} characteristic and mean Betti's
numbers in the Ising model with arbitrary spin on $\mathbbm{Z}^2$ as functions
of the temperature is investigated through intensive Monte Carlo simulations.
We also consider these quantities for each color $a$ in the state space $S\_Q =
\{- Q, - Q + 2, ..., Q \}$ of the model. We find that these topological
invariants show a sharp transition at the critical point.
