  Given a Poisson process on a $d$-dimensional torus, its random geometric
simplicial complex is the complex whose vertices are the points of the Poisson
process and simplices are given by the \u{C}ech complex associated to the
coverage of each point. By means of Malliavin calculus, we compute explicitly
the n$th$ order moment of the number of $k$-simplices. The two first order
moments of this quantity allow us to find the mean and the variance of the
Euler caracteristic. Also, we show that the number of any connected geometric
simplicial complex converges to the Gaussian law when the intensity of the
Poisson point process tends to infinity. We use a concentration inequality to
find bounds for the for the distribution of the Betti number of first order and
the Euler characteristic in such simplicial complex.
