The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds

In a recent work (Bagchi and Datta, 2014) with Datta, we introduced the mu-vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu-vectors. With the new definition, most results of Bagchi and Datta (2014) become correct without the hypothesis of 2-neighbourliness. In particular, the combinatorial Morse inequalities of Bagchi and Datta (2014) are now true of all simplicial complexes.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 d, then for 1≤ℓ≤d−12 and any field F,gℓ+1(M)≥(d+2ℓ+1)∑i=1ℓ(−1)ℓ−iβi(M;F). Equality holds here if and only if M is ℓ-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)≥10β1(M;F), with equality iff M is 1-stacked. This result settles a question of Novik and Swartz (2009) in the affirmative.