An infinite family of tight triangulations of manifolds

TL;DR

This paper presents a new explicit construction of infinite families of tight, vertex-transitive triangulations of d-manifolds with minimal vertices, expanding the known examples in geometric topology.

Contribution

The authors construct two infinite series of tight, neighborly triangulated d-manifolds with minimal vertices, extending the class of known tight triangulations beyond K"uhnel's examples.

Findings

Constructed explicit vertex-transitive tight triangulations for all d ≥ 2.

Manifolds are strongly minimal and have stacked sphere vertex-links.

Results include orientability properties depending on dimension.

Abstract

We give an explicit construction of vertex-transitive tight triangulations of $d$-manifolds for $d\geq 2$. More explicitly, for each $d\geq 2$, we construct two $(d^2+5d+5)$-vertex neighborly triangulated $d$-manifolds whose vertex-links are stacked spheres. The only other non-trivial series of such tight triangulated manifolds currently known is the series of non-simply connected triangulated $d$-manifolds with $2d+3$ vertices constructed by K\"{u}hnel. The manifolds we construct are strongly minimal. For $d\geq 3$, they are also tight neighborly as defined by Lutz, Sulanke and Swartz. Like K\"{u}hnel's complexes, our manifolds are orientable in even dimensions and non-orientable in odd dimensions.