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

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 $\mathcal{G}_M \geq 2\chi(M) + 5m + 11$. Moreover, a 4-manifold $M$ is PL homeomorphic to the 4-sphere if and only if its balanced genus satisfies $\mathcal{G}_M \leq 2\chi(M) + 10$. 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.