  A simplicial complex $X$ is said to be tight with respect to a field
$\mathbb{F}$ if $X$ is connected and, for every induced subcomplex $Y$ of $X$,
the linear map $H_\ast (Y; \mathbb{F}) \rightarrow H_\ast (X; \mathbb{F})$
(induced by the inclusion map) is injective. This notion was introduced by
K\"{u}hnel in [10]. In this paper we prove the following two combinatorial
criteria for tightness. (a) Any $(k+1)$-neighbourly $k$-stacked
$\mathbb{F}$-homology manifold with boundary is $\mathbb{F}$-tight. Also, (b)
any $\mathbb{F}$-orientable $(k+1)$-neighbourly $k$-stacked
$\mathbb{F}$-homology manifold without boundary is $\mathbb{F}$-tight, at least
if its dimension is not equal to $2k+1$.
  The result (a) appears to be the first criterion to be found for tightness of
(homology) manifolds with boundary. Since every $(k+1)$-neighbourly $k$-stacked
manifold without boundary is, by definition, the boundary of a
$(k+1)$-neighbourly $k$-stacked manifold with boundary - and since we now know
several examples (including two infinite families) of triangulations from the
former class - theorem (a) provides us with many examples of tight triangulated
manifolds with boundary.
  The second result (b) generalizes a similar result from [2] which was proved
for a class of combinatorial manifolds without boundary. We believe that
theorem (b) is valid for dimension $2k+1$ as well. Except for this lacuna, this
result answers a recent question of Effenberger [8] affirmatively.
