A characterization of homology manifolds with $g_2\leq 2$

TL;DR

This paper characterizes homology manifolds with low $g_2$ values, extending known results and showing that all such manifolds with $g_2=2$ are polytopal spheres, using retriangulation techniques.

Contribution

It provides a new proof for the characterization of homology spheres with $g_2=1$ and extends the classification to manifolds with $g_2=2$, demonstrating they are polytopal spheres.

Findings

Homology $(d-1)$-spheres with $g_2=1$ are characterized for $d extgreater=5$.

Homology manifolds with $g_2=2$ are obtained by centrally retriangulating polytopal spheres with $g_2 extless=1$.

All homology manifolds with $g_2=2$ are polytopal spheres.

Abstract

We characterize homology manifolds with $g_2\leq 2$. Specifically, using retriangulations of simplicial complexes, we give a short proof of Nevo and Novinsky's result on the characterization of homology $(d-1)$-spheres with $g_2=1$ for $d\geq 5$ and extend it to the class of normal pseudomanifolds. We proceed to prove that every prime homology manifold with $g_2=2$ is obtained by centrally retriangulating a polytopal sphere with $g_2\leq 1$ along a certain subcomplex. This implies that all homology manifolds with $g_2=2$ are polytopal spheres.