Bistellar Equivalences of two Families of Simplicial Complexes

Abstract

In this paper, we study a pair of simplicial complexes, which we denote by $\mathcal{B}(k,d)$ and $\mathcal{S}\mathcal{T}(k+1,d-k-1)$, for all nonnegative integers $k$ and $d$ with $0\le k\le d-2$. We conjecture that their underlying topological spaces $|\mathcal{B}(k,d)|$ and $|\mathcal{S}\mathcal{T}(k+1,d-k-1)|$ are homeomorphic for all such $k$ and $d$. We answer this question when $k=d-2$ by relating the complexes through a series of well-studied combinatorial operations that transform a combinatorial manifold while preserving its PL-homeomorphism type.