On the mean Euler characteristic and mean Betti numbers of the Ising model with arbitrary spin

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The behaviour of the mean Euler-Poincaré characteristic and mean Betti's numbers in the Ising model with arbitrary spin on Z² 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.