  An exact expression for the Green function of a purely fermionic system
moving on the manifold $\Re \times \Sigma^{D-1}$, where $\Sigma^{D-1}$ is a
$(D-1)$-torus, is found. This expression involves the bosonic analog of $\chi_n
= e^{in\theta}$ corresponding to the irreducible representation for the n-th
class of homotopy and in the fermionic case for D=2 and 3, $\chi_n$ is a
measure of the statistics of the particles. For higher dimensions ($D \geq 4$),
there is no analogue interpretation however this could, presumably, indicate a
generation of mass as in quantum field theories at finite temperature.
