Embeddings of edge-colored dual graphs of balanced 3- and 4-manifolds

TL;DR

The paper introduces the balanced genus as a new invariant arising from the edge-colored dual graphs of balanced normal pseudomanifolds, linking combinatorics of colored triangulations to surface embeddings. It establishes lower bounds for the balanced genus in 3- and 4-manifolds in terms of fundamental-group rank and Euler characteristic, and provides sphere-recognition criteria: a 3-manifold is S^3 iff 𝒢_M ≤ 3, and a 4-manifold is S^4 iff 𝒢_M ≤ 2χ(M)+10. The results demonstrate sharpness for certain sphere bundles and lay out a program of future work, including classification by balanced genus, higher-dimensional generalizations, and links to normal pseudomanifolds and algebraic combinatorics. The framework blends dual-graph embeddings, edge-path groups, and flag-vector refinements to connect PL topology with combinatorial invariants.

Abstract

This article focuses on a class of properly edge-colored graphs, which arise from topological combinatorics, and investigates their embeddings onto surfaces. Specifically, these graphs are known as the dual graphs of balanced normal pseudomanifolds. We introduce the concept of the balanced genus, which represents the smallest genus of a surface onto which the dual graph of a normal pseudomanifold can embed regularly. As a key result, we establish that for any 3-manifold $ M $ that is not a sphere, the balanced genus satisfies the lower bound $ \mathcal{G}_M \geq m+3 $, where $ m $ is the rank of its fundamental group of $M$. Furthermore, we prove that a 3-manifold $ M $ is homeomorphic to the 3-sphere if and only if its balanced genus $ \mathcal{G}_M $ is at most 3. Similarly, for 4-manifolds, we establish that if $ M $ is not homeomorphic to a sphere, then its balanced genus is bounded below by $ \mathcal{G}_M \geq 2χ(M) + 5m + 11 $. Moreover, a 4-manifold $ M $ is PL homeomorphic to the 4-sphere if and only if its balanced genus satisfies $ \mathcal{G}_M \leq 2χ(M) + 10 $. We believe that the balanced genus offers a new perspective in graph theory and combinatorics and will inspire further developments in the field in connection with algebraic combinatorics. To this end, we outline several directions for future research.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• Corollary 2.3
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof