A structure theorem for homology 4-manifolds with $g_2\le 5$

Numerous structural findings of homology manifolds have been derived in various ways in relation to $g_2$-values. The homology $4$-manifolds with $g_2\le 5$ are characterized combinatorially in this article. It is well-known that all homology $4$-manifolds for $g_2\le 2$ are polytopal spheres. We demonstrate that homology $4$-manifolds with $g_2\le 5$ are triangulated spheres and are derived from triangulated 4-spheres with $g_2\le 2$ by a series of connected sum, bistellar 1- and 2-moves, edge contraction, edge expansion, and edge flipping operations. We establish that the above inequality is optimally attainable, i.e., it cannot be extended to $g_2=6$.