The Lower Bound Conjecture for 3- and 4-manifolds

For any closed connected d-manifold M let ](M) denote the set of vectors / (K)= (]0(K) ..... fd(K)), where K ranges over all triangulations of M and ]k(K) denotes the number of k-simplices of K. The principal results of this paper are Theorems 1 through 5 below, which, together with the Dehn-Sommerville equations reviewed in w 2, yield a characterization of ](M) for some of the simpler 3and 4-manifolds. The results for the 3and 4spheres given in Theorems 1 and 5 have immediate and obvious implications for simplicial polytopes, i.e., closed bounded convex polyhedra all of whose proper faces are simplices. In particular they provide a strong affirmative resolution in dimensions 4 and 5 of the socalled lower bound conjecture for simplicial polytopes. For a discussion of this conjecture, which in dimension 4 goes back at least to a paper by Briiclmer in 1909, and some limited results in higher dimensions the reader is referred to Section 10.3 of Griinbaum's book on polytopes [2]. Theorem 3, which is concerned with triangulations of projective 3-space, also has an immediate implication for a special subclass of the centrally symmetric simplicial polytopes. This result is stated as Theorem 6. Some special classes ~/d(n), d >~ 1, n/>0, of abstract simplicial complexes figure in the statement and proof of these theorems. For d >~ 2 each class ~,td(n) consists of certain especially simple triangulations of a class of closed d-manifolds which might be described as d-spheres with n orientable or nonorientable handles. The classes ~/d(n) may be defined inductively as follows: