  In a recent work [2] 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 [2] become correct without the hypothesis of
2-neighbourliness. In particular, the combinatorial Morse inequalities of [2]
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 $1 \leq \ell \leq \frac{d-1}{2}$ and any
field $\mathbb{F}, ~ g_{\ell+1} (M) \geq \binom{d+2}{\ell+1}
\sum\limits_{i=1}^\ell (-1)^{\ell-i} \beta_i (M;\mathbb{F})$. Equality holds
here if and only if $M$ is $\ell$-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 GLBC for all connected and
closed combinatorial 3-manifolds. Thus, any connected closed combinatorial
manifold $M$ of dimension three satisfies $g_2 (M) \geq 10 \beta_1
(M;\mathbb{F})$, with equality iff $M$ is 1-stacked. This result settles a
question of Novik and Swartz [6] in the affirmative.
