A characterization of homology manifolds with g2 ≤ 2

We characterize homology manifolds with g2≤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 g2=1 for d≥5 and extend it to the class of normal pseudomanifolds. We proceed to prove that every prime homology manifold with g2=2 is obtained by centrally retriangulating a polytopal sphere with g2≤1 along a certain subcomplex. This implies that all homology manifolds with g2=2 are polytopal spheres.