  As shown by early studies on mean-field models of the glass transition, the
geometrical features of the energy landscape provide fundamental information on
the dynamical transition at the Mode-Coupling temperature $T_d$. We show that
active particles can serve as a useful tool for gaining insight into the
topological crossover in model glass-formers. In such systems the landmark of
the minima-to-saddle transition in the potential energy landscape, taking place
in the proximity of $T_d$, is the critical slowing down of dynamics.
Nevertheless, the critical slowing down is a bottleneck for numerical
simulations and the possibility to take advantage of the new smart algorithms
capable to thermalize down in the glass phase is attractive. Our proposal is to
consider configurations equilibrated below the threshold and study their
dynamics in the presence of a small amount of self-propulsion. As exemplified
here from the study of the p-spin model, the presence of self-propulsion gives
rise to critical off-equilibrium equal-time correlations at the
minima-to-saddles crossover, correlations which are not hindered by the
sluggish glassy dynamics.
