The generic combinatorial simplex

TL;DR

This paper introduces the generic combinatorial n-simplex using projective Fra"issé theory, providing a new combinatorial framework for understanding simplices and their geometric realizations without Euclidean reference.

Contribution

It defines the generic combinatorial n-simplex via projective Fra"issé theory and explores its properties, including domination closure and relations to cellular maps and near-homeomorphisms.

Findings

The generic combinatorial n-simplex is canonically associated with finite triangulations.

Domination closure of selection maps includes face-preserving cellular maps.

Under the PL-Poincaré conjecture, it characterizes the domination closure of selections.

Abstract

We employ projective Fra\"iss\'e theory to define the "generic combinatorial $n$-simplex" as the pro-finite, simplicial complex that is canonically associated with a family of simply defined selection maps between finite triangulations of the simplex. The generic combinatorial $n$-simplex is a combinatorial object that can be used to define the geometric realization of a simplicial complex without any reference to the Euclidean space. It also reflects dynamical properties of its homeomorphism group down to finite combinatorics. As part of our study of the generic combinatorial simplex, we define and prove results on domination closure for Fra\"iss\'e classes, and we develop further the theories of stellar moves and cellular maps. We prove that the domination closure of selection maps contains the class of face-preserving simplicial maps that are cellular on each face of the $n$-simplex and is contained in the class of simplicial, face-preserving near-homeomorphisms. Under the PL-Poincar\'e conjecture, this gives a characterization of the domination closure of selections.