A tightness criterion for homology manifolds with or without boundary

TL;DR

This paper establishes new combinatorial criteria for tightness in homology manifolds, providing the first such criteria for manifolds with boundary and extending known results to boundaryless cases.

Contribution

It introduces the first criteria for tightness of homology manifolds with boundary and generalizes existing results to broader classes of boundaryless manifolds.

Findings

Any $(k+1)$-neighbourly $k$-stacked $ ext{F}$-homology manifold with boundary is $ ext{F}$-tight.

Any $ ext{F}$-orientable $(k+1)$-neighbourly $k$-stacked $ ext{F}$-homology manifold without boundary (not of dimension $2k+1$) is $ ext{F}$-tight.

Provides numerous examples of tight triangulated manifolds with 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.