An Upper and a Lower Bound Theorem for Combinatorial 4-Manifolds

In this paper we prove two inequalities. The first one gives a lower bound for the Euler characteristic of a tight combinatorial 4-manifold M under the additional assumptions that |M| is 1-connected, that M is a subcomplex of H (M) , and that H (M) is a centrally symmetric and simplicial d -polytope. The second inequality relates the Euler characteristic with the number of vertices of a combinatorial 4-manifold admitting a fixed-point free involution. Furthermore, we construct a new and highly symmetric 12-vertex triangulation of S ² x S ² realizing equality in each of these inequalities. Received January 23, 1996, and in revised form September 13, 1996.