As an application, we prove the following generalized lower bound theorem (GLBT) for connected locally tame combinatorial manifolds. If M is such a manifold of dimension 18730e17a68f7eae9b6126" title="Click to view the MathML source">d, then for and any field . Equality holds here if and only if M is 鈩?/span>-stacked.
We conjecture that, more generally, this theorem is true of all triangulated connected and closed homology manifolds. A conjecture on the sigma-vectors of triangulated homology spheres is proposed, whose validity will imply this GLB Conjecture for homology manifolds. We also prove the GLB Conjecture for all connected and closed combinatorial 3-manifolds. Thus, any connected closed combinatorial manifold M of dimension three satisfies g2(M)≥10尾1(M;F), with equality iff M is 1-stacked. This result settles a question of Novik and Swartz (2009) in the affirmative.