The celebrated low bound theorem states that any simplicial manifold of dimension ≥ 3 satisﬁes \$g_2 \geq 0\$, and equality holds if and only if it is a stacked sphere. Furthermore, more recently, the class of all simplicial spheres with \$g_2 = 1\$ was characterized by Nevo and Novinsky, by an argument based on rigidity theory for graphs. In this talk, I will ﬁrst deﬁne three diﬀerent retriangulations of simplicial complexes that preserve the homeomorphism type. Then I will show that all simplicial manifolds with \$g_2 \leq 2\$ can be obtained by retriangulating a polytopal sphere with a smaller \$g_2\$. This implies Nevo and Novinskys result for simplicial spheres of dimension \$\geq 4\$. More surprisingly, it also implies that all simplicial manifolds with \$g_2 = 2\$ are polytopal spheres.