Resorting to microcanonical ensemble Monte Carlo simulations, we study the geometric 1 and topological properties of the state space of a model of network glass-former. This model, a 2 Lennard-Jones binary mixture, does not crystallize due to frustration. We found, at equilibrium and 3 at low energy, two peaks of specific heat, in correspondence with important changes of local ordering. 4 These singularities are accompanied by inflection points of geometrical markers of the potential 5 energy level sets, namely the mean curvature, the dispersion of the principal curvatures and the 6 variance of the scalar curvature. Pinkall’s and Overholt’s theorems closely relate these quantities to 7 the topological properties of the accessible state-space manifold. Thus, our analysis provides strong 8 indications that the glass transition is associated with major changes of the topology of the energy 9 level sets. This important result suggests that this phase transition can be understood through the 10 topological theory of phase transitions.