The Lower Bound Conjecture for 3- and 4-Manifolds

摘要

The so-called lower bound conjecture for simplicial polytopes asserts that e(P) = or > d.v(P) - d(d+1)/2, where e(P) and v(P) denote respectively the number of edges and vertices of any simplicial d-polytope P, i.e., any closed bounded convex polyhedron of dimension d, all of whose faces are simplices. This paper establishes analogous lower bounds for arbitrary triangulations of closed topological 3- and 4-manifolds, including sharp lower bounds for the 3-sphere, the 3-dimensional analogues of the torus and Klein bottle, projective 3-space, and the 4-sphere with any number of handles. The results for the 3-sphere and 4-sphere immediately imply the previously unproven lower bound conjecture for simplicial 4- and 5-polytopes. The result for projective 3-space has similar implications for centrally symmetric simplicial 4-polytopes. (Author)