Simplicial maps between spheres and Davis' manifolds with positive simplicial volume

TL;DR

The paper studies when Davis' manifolds $M(T)$, built from sphere triangulations $T$, have positive simplicial volume and introduces a dominance order $T_1\\preceq T_2$ based on nonzero-degree simplicial maps. It proves a complete 3D picture: for flag $T$ on $S^2$, $\\norm M(T) \\>0$ iff $T_9 \\preceq T$, with a degree-1 map $T\\to T_9$, so $T_9$ is the unique minimal positive example; this result leverages relatively hyperbolic group theory and graph-minor techniques. In dimension 4, two basic flag triangulations $T_{10}$ and $T_{12}$ arise with nonzero Euler characteristic, with $M(T_{10})$ known to have positive simplicial volume (as a product of genus-5 surfaces), while the status of $M(T_{12})$ remains open; extensive computer experiments and the novel local-picture framework are developed to search for further basics. The work connects dominance to graph-minor theory, discusses algorithmic aspects for testing domination, and outlines open problems about higher-dimensional cases and broader classes of Davis' manifolds. Overall, it advances a combinatorial-geometry program for understanding when aspherical Davis' manifolds have positive simplicial volume via minimal sphere triangulations and their maps.

Abstract

We study the simplicial volume of manifolds obtained from Davis' reflection group trick, the goal being characterizing those having positive simplicial volume. In particular, we focus on checking whether manifolds in this class with nonzero Euler characteristic have positive simplicial volume (Gromov asked whether this holds in general for aspherical manifolds). This leads to a combinatorial problem about triangulations of spheres: we define a partial order on the set of triangulations -- the relation being the existence of a nonzero-degree simplicial map between two triangulations -- and the problem is to find the minimal elements of a specific subposet. We solve explicitly the case of triangulations of the two-dimensional sphere, and then perform an extensive analysis, with the help of computer searches, of the three-dimensional case. Moreover, we present a connection of this problem with the theory of graph minors.

Figures (18)

• Figure 1: The $T_9$ triangulation of $S^2$.
• Figure 2: The octahedral triangulation of $S^2$.
• Figure 3: The subcomplex to be replaced.
• Figure 4: The subcomplex that replaces the previous one.
• Figure 5: The collapse of an edge.

Theorems & Definitions (84)

• Definition 1.2
• Theorem 1.3: \ref{['thm:2dim']}
• Proposition 1.4: \ref{['prop:minor_s2']}
• Proposition 1.5: \ref{['cor:dim2_deg']}
• Proposition 1.6
• Proposition 1.8: \ref{['prop:many_squares']}
• Proposition 2.1: BH2013
• Definition 2.2
• Definition 2.3: Edge subdivision
• Definition 2.4: Edge collapse