A Construction Principle for Tight and Minimal Triangulations of Manifolds

ABSTRACT

Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold. Tight triangulations are conjectured to be strongly minimal and proven to be so for dimensions ⩽ 3. However, in spite of substantial theoretical results about such triangulations, there are precious few examples. In fact, apart from dimension two, we do not know if there are infinitely many of them in any given dimension. In this article, we present a computer-friendly combinatorial scheme to obtain tight triangulations and present new examples in dimensions three, four, and five. Furthermore, we describe a family of tight triangulated d-manifolds, with 2^{d − 1}⌊d/2⌋!⌊(d − 1)/2⌋! isomorphically distinct members for each dimension d ⩾ 2. While we still do not know if there are infinitely many tight triangulations in a fixed dimension d > 2, this result shows that there are abundantly many.

KEYWORDS:

• combinatorial manifold
• (embeddings of) abstract simplicial complexes
• stacked sphere
• strongly minimal triangulation
• tight triangulation

2000 AMS SUBJECT CLASSIFICATION:

• 57Q15
• 57R05

Acknowledgments

The authors thank the anonymous referees for useful comments.

Additional information

Funding

This work is supported by DIICCSRTE, Australia (project AISRF06660) & DST, India (DST/INT/AUS/P-56/2013 (G)) under the Australia–India Strategic Research Fund. The second author is also supported by the UGC Centre for Advanced Studies.