Vertex-Minimal Triangulation of Complexes with Homology

TL;DR

This paper determines the minimal number of vertices needed for pure d-dimensional simplicial complexes with non-trivial homology in a given dimension, also considering strong connectivity constraints.

Contribution

It provides exact minimal vertex counts for complexes with specified homology and connectivity, advancing understanding of the combinatorial topology of such complexes.

Findings

Minimal vertices for complexes with non-trivial homology established

Solutions under strong connectivity constraints provided

Results applicable to combinatorial and algebraic topology

Abstract

For a given pair of numbers $(d,k)$, we establish the minimal number of vertices in pure $d$-dimensional simplicial complexes with non-trivial homology in dimension $k$. Furthermore, we solve the problem under the additional constraint of strong connectivity with respect to any intermediate dimension.