On a construction of some homology $d$-manifolds

The $g$-vector of a simplicial complex contains a lot of information about the combinatorial and topological structure of that complex. Several classification results regarding the structure of normal pseudomanifolds and homology manifolds have been established concerning the value of $g_2$. It is known that when $g_2=0$, all normal pseudomanifolds of dimensions at least three are stacked spheres. In the cases of $g_2=1$ and $2$, all homology manifolds are polytopal spheres and can be obtained through retriangulation or join operations from the previous ones. In this article, we provide a combinatorial characterization of the homology $d$-manifolds, where $d\geq 3$ and $g_2=3$. These are spheres and can be obtained through operations such as joins, some retriangulations, and connected sums from spheres with $g_2\leq 2$. Furthermore, we have presented a structural result on prime normal $d$-pseudomanifolds with $g_2=3$.