Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Euler's Theorem for Regular CW-Complexes

Richard H. Hammack, Paul C. Kainen

TL;DR

The paper generalizes Euler's classic equivalences from connected multigraphs to pure, strongly connected $n$-complexes by introducing the evenness condition $\deg_K(s)$ even for all $(n-1)$-cells $s$ and two higher-dimensional analogues: the $n$-circlet and the $n$-Euler cover. It proves that for a pure, $s$-connected $n$-complex $K$, (i) $K$ is even, (ii) $K$ can be decomposed as a facet-disjoint union of circlets, and (iii) there exists an $n$-pseudomanifold $M$ with an Euler cover $\varphi: M \to K$; the equivalence is established via an explicit construction that duplicates codimension-1 data and reattaches facets through a gluing mechanism. Central to the argument is the notion that every circlet admits an Euler cover and that the global structure can be assembled from circlets using recombination, yielding a higher-dimensional generalization of cycle decompositions and Eulerian traversals. By leveraging dual graphs, pseudomanifold properties, and a detailed gluing framework, the results extend prior 2D theories and intersect recent hypergraph Eulerian results, offering a topology-inspired pathway to decompositions and traversals in higher dimensions with potential applications in topology-driven data analysis and robotics.

Abstract

For strongly connected, pure $n$-dimensional regular CW-complexes, we show that {\it evenness} (each $(n{-}1)$-cell is contained in an even number of $n$-cells) is equivalent to generalizations of both cycle decomposition and traversability.

Euler's Theorem for Regular CW-Complexes

TL;DR

The paper generalizes Euler's classic equivalences from connected multigraphs to pure, strongly connected -complexes by introducing the evenness condition even for all -cells and two higher-dimensional analogues: the -circlet and the -Euler cover. It proves that for a pure, -connected -complex , (i) is even, (ii) can be decomposed as a facet-disjoint union of circlets, and (iii) there exists an -pseudomanifold with an Euler cover ; the equivalence is established via an explicit construction that duplicates codimension-1 data and reattaches facets through a gluing mechanism. Central to the argument is the notion that every circlet admits an Euler cover and that the global structure can be assembled from circlets using recombination, yielding a higher-dimensional generalization of cycle decompositions and Eulerian traversals. By leveraging dual graphs, pseudomanifold properties, and a detailed gluing framework, the results extend prior 2D theories and intersect recent hypergraph Eulerian results, offering a topology-inspired pathway to decompositions and traversals in higher dimensions with potential applications in topology-driven data analysis and robotics.

Abstract

For strongly connected, pure -dimensional regular CW-complexes, we show that {\it evenness} (each -cell is contained in an even number of -cells) is equivalent to generalizations of both cycle decomposition and traversability.
Paper Structure (6 sections, 7 theorems, 7 equations, 3 figures)

This paper contains 6 sections, 7 theorems, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and main result
  3. Some properties of circlets and pseudomanifolds
  4. Machinery and proof of main theorem
  5. Historical Context
  6. Discussion

Key Result

Lemma 1

For $K$ an even pure $n$-complex, $K$ is a circlet $\iff$$K$ is minimal even.

Figures (3)

  • Figure 1: A family of circlets $C(k,m)$. Let $k \geq 3$ and let $m \geq 4$ be even. To construct the 2-circlet $C(k,m)$, begin with the 2-complex illustrated here, whose 1-skeleton is the graph Cartesian product $K_{1,m}\Box P_{k+1}$. Identify the left $K_{1,m}$ with the right $K_{1,m}$ by a twist of $2\pi/m$, as indicated. The result is a finned 2-complex with a $C_{km}$ boundary. Cap this boundary by a polygon with $km$ sides. The result is a 2-circlet, as the $km$-gon shares an edge of degree 2 with any square, so any even $n$-subcompex that contains the $km$-gon necessarily contains all the squares too.
  • Figure 2: An Euler cover of an even 1-complex $K$. A vertex labeled $x$ in $M$ maps to $x$ in $K$. An edge labeled $xy$ in $M$ maps to $xy$ in $K$.
  • Figure 3: An Euler cover $\varphi\colon M\to \Delta_5^{2}$ of the 2-skeleton of the 5-dimensional simplex $\Delta_5$, which has six vertices $1,2,3,4,5,6$, fifteen edges and twenty triangular faces. The pseudomanifold $M$ has six vertices, thirty edges and twenty triangular faces (including the unbounded face in this planar drawing). Vertices with the same labels are identified, so $M$ is a sphere with six pinchpoints. Note that the three shaded areas of $M$ map to three pairwise face-disjoint tetrahedron boundaries in $K$. The white triangles (including the unbounded region) map to the boundary of an octahedron in $K$. This decomposes $K$ as a face-disjoint union of four spheres (circlets).

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 1
  • ...and 4 more