The celebrated lower bound theorem states that any simplicial manifold of dimension $\ge3$ satisfies $g_2\ge0$, 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 $g_2\le2$ can be obtained by retriangulating a polytopal sphere with a smaller $g_2$. This implies Nevo and Novinsky's result for simplicial spheres of dimension $\ge4$. More surprisingly, it also implies that all simplicial manifolds with $g_2=2$ are polytopal spheres.