Face numbers of triangulations of manifolds

Abstract

In this paper we discuss face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of triangulations of simply connected $4$-manifolds: We observe that, for a triangulation $\mathcal{T}$ of a simply connected $4$-manifold $\mathcal{M}$ with $n$ pentachora, an upper bound on the number of vertices $v$ of $\mathcal{T}$ as a function of $n$ yields a lower bound for $n$ depending only on the second Betti number $β_2(\mathcal{M})$ of $\mathcal{M}$. Within this framework we conjecture that $v \leq \frac{n}{2}+4$, implying $n \geq 2β_2(\mathcal{M})$. In forthcoming work by the authors, this conjectured bound is shown to be almost tight for all values of $β_2(\mathcal{M})$, with a gap of at most two. We extend our conjecture to arbitrary dimensions and show that an $n$-facet triangulation of an odd-dimensional $d$-manifold, $n \geq d$, can have at most $n + \frac{d-1}{2}$ vertices, and conjecture that, for $d$ even, the bound is $\frac{n}{2}+d$. We show that these (conjectured) bounds are (would be) tight for all odd (even) dimensions and all values of $n \geq d$. Finally, we give necessary conditions for the dual graph of $\mathcal{T}$ to satisfy our conjecture. We furthermore present families of $4$-dimensional pseudomanifolds with singularities in their edge links that have more than $\frac{n}{2}+4$ vertices, thereby proving that the manifold condition is necessary for our conjecture to hold.

Figures (6)

• Figure 3.1: Dual graphs of triangulations obtained by repeated $0$-$2$ vertex moves on the pillow triangulation of $\mathbb{S}^4$ (top left) with up to 8 pentachora. By construction, these all achieve $\delta_{\mathcal{T}} = f_0 - \frac{f_4}{2} = 4$.
• Figure 4.1: An example of a graph (left) and its block-separating tree (right). Separating nodes are highlighted in red in both. The black nodes of the block-separating tree correspond to non-separable components of the original graph.
• Figure 4.2: Replacing a loop with a "ringpull" via a $0$-$2$ vertex move, in dimension 4.
• Figure 4.3: Example of a 5-regular graph $\Gamma$ arranged as an "antenna", with nodes ordered as in the proof of \ref{['lem:branch-equals-crit']}. Arcs belonging to each non-separable component are highlighted alternately in red, blue or green. Observe that each is arranged horizontally or vertically according to a sequence with two critical points. Since $\Gamma$ has loops, our sequence with $\mathop{\mathrm{branch}}\nolimits(\Gamma)$ critical points first requires expanding them as in \ref{['fig:loop-removal']}. The antenna has $13=\mathop{\mathrm{branch}}\nolimits(\Gamma)$ "spikes" (noting loops count as spikes, unless they extend an existing spike like the loop above $v_{26}$), and the order of the nodes has seven critical points. After expanding the loops, six more critical points are needed.
• Figure 5.1: Dual graphs of the triangulations (left to right) $\mathbb{S}^5_5$, $\mathbb{S}^5_6$ and $\mathbb{S}^5_7$ as constructed in \ref{['prop:construction-odd-large']}.

Theorems & Definitions (53)

• Definition 2.1
• Theorem 2.2: see Burton2012-Regina
• Remark 3.1
• Lemma 3.2
• proof
• Remark 3.3
• Proposition 3.4
• proof
• Corollary 3.5
• proof