Three-dimensional normal pseudomanifolds with relatively few edges

Abstract

Let Δ be a d-dimensional normal pseudomanifold, d≥3. A relative lower bound for the number of edges in Δ is that g2 of Δ is at least g2 of the link of any vertex. When this inequality is sharp Δ has relatively minimalg2. For example, whenever the one-skeleton of Δ equals the one-skeleton of the star of a vertex, then Δ has relatively minimal g2. Subdividing a facet in such an example also gives a complex with relatively minimal g2. We prove that in dimension three these are the only examples. As an application we determine the combinatorial and topological type of 3-dimensional Δ with relatively minimal g2 whenever Δ has two or fewer singularities. The topological type of any such complex is a pseudocompression body, a pseudomanifold version of a compression body.Complete combinatorial descriptions of Δ with g2(Δ)≤2 are due to Kalai [12](g2=0), Nevo and Novinsky [13](g2=1) and Zheng [20](g2=2). In all three cases Δ is the boundary of a simplicial polytope. Zheng observed that for all d≥0 there are triangulations of Sd⁎RP2 with g2=3. She asked if this is the only nonspherical topology possible for g2(Δ)=3. As another application of relatively minimal g2 we give an affirmative answer when Δ is 3-dimensional.