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Register a new userA tightness criterion for homology manifolds with or without boundary
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Bhaskar Bagchi
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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 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.
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