The Face Group of a Simplicial Complex

TL;DR

This work defines the face group $F(X,x_0)$ as a purely combinatorial analogue of the second homotopy group, built from face spheres $I_m\times I_n \to X$ and organized via extension-contiguity. It develops a robust apparatus—contiguity, trivial extensions, and a careful treatment of rectangular domains—that yields a well-defined group structure and functoriality, with $F(X,x_0)$ behaving well under products. The central result establishes an isomorphism $F(X,x_0)\cong\pi_2(|X|,x_0)$, achieved by connecting combinatorial maps to spatial realizations through a detailed simplicial-approximation framework that avoids barycentric subdivision. An intrinsic description mirrors the edge group while capturing higher-dimensional data, and the framework is poised to extend to higher $\pi_k$ with potential applications in digital topology and combinatorial loop spaces.

Abstract

The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalence classes of edge loops and that is isomorphic to the ordinary (topological) fundamental group of the spatial realization. We define a counterpart to the edge group that likewise gives a combinatorial version of the second (higher) homotopy group. Working entirely combinatorially, we show our group is an abelian group and also respects products. We show that our combinatorially defined group is isomorphic to the ordinary (topological) second homotopy group of the spatial realization.

Figures (14)

• Figure 1: $I_5 \times I_4$ represented as the clique complex of the graph $G_{5, 4}$.
• Figure 2: The $2$-sphere as a simplicial complex $X$.
• Figure 3: A face sphere in $X$, in this case a simplicial map $f: \left(I_5 \times I_4, \partial (I_5 \times I_4) \right) \to (S^2,-\mathbf{e}_1)$
• Figure 4: Schematic of the product $f\cdot g$ of two maps $f$ and $g$.
• Figure 5: Values of $g_{r}$ on the vertices of $[r-1, r+1]\times [j, j+1]$, with $x_{i, j}$ denoting $f(i, j)$.

Theorems & Definitions (64)

• Theorem
• Remark 2.1
• Example 2.2
• Lemma 3.1
• proof
• Definition 3.2
• Proposition 3.3
• proof
• Definition 3.4
• Proposition 3.5