A tightness criterion for homology manifolds with or without boundary

Abstract

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.