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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 byK\"{u}hnel in [10]. In this paper we prove the following two combinatorialcriteria 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 leastif 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 $k$-stackedmanifold without boundary is, by definition, the boundary of a$(k+1)$-neighbourly $k$-stacked manifold with boundary - and since we now knowseveral examples (including two infinite families) of triangulations from theformer class - theorem (a) provides us with many examples of tight triangulatedmanifolds with boundary.  The second result (b) generalizes a similar result from [2] which was provedfor a class of combinatorial manifolds without boundary. We believe thattheorem (b) is valid for dimension $2k+1$ as well. Except for this lacuna, thisresult answers a recent question of Effenberger [8] affirmatively.

A tightness criterion for homology manifolds with or without boundary