On the Fundamental Group of 3-Gems and a ‘Planar’ Class of 3-manifolds

Abstract

In this paper we associate a group γ H to each bipartite 3-gem H . By using the recent (Theorem 1) graph-theoretical characterization of homeomorphisms among closed 3-manifolds due to Ferri and Gagliardi [6] this group is proven to be an invariant of |H|, the closed 3-manifold associated to H . The definition of γ H and the proof of the invariance are entirely combinatorial but a direct geometric interpretation (via Heegaard diagrams) shows that γ H is the fundamental group of |H|. Theorem 2 is, then, a generalization for the case of 3-dimensional orientable closed manifolds of results in [7] and [2]. In the final section we show how to construct a 3-gem ψ( G ) from a planar graph G . By using theorems 2 and 3 we prove that the construction ψ has the following property: the number of spanning trees of G is equal to the order of the first homology group of |ψ( G )|. We also show that ψ( G ) = ψ( H ) if G and H are geometric duals [26].