Detection of first homology via random geometric graphs in the thermodynamic regime

Consider a random geometric graph $G_M(n;r)$ on a compact Riemannian manifold $M$, whose vertices are a cloud of $n$ independently sampled points, and whose edges connect vertices at distance $\le r$. We show that, in the thermodynamic (i.e. bounded expected average degree) regime, if $G_M(n;r)$ i