# A combinatorial proof of the Lower Bound Theorem

An m-dimensional triangulated ball B is called a stacked ball if there is a sequence B1, . . . , Bk of triangulated m-balls such that B1 is a simplex, Bk = B and Bj+1 can be constructed from Bj by attaching an m-simplex along an (m−1)-face of Bj for 1 ≤ j ≤ k−1. The boundary complex of a stacked (d + 1)-ball is called a stacked d-sphere. The lower bound theorem (LBT) says that the number of edges of a closed, connected, n-vertex triangulated d-manifold is at least that of an n-vertex stacked d-sphere and equality attains only for stacked d-sphere for d ≥ 3. So, the LBT provides the best possible lower bound for the number of edges for any triangulated manifold. In this talk, we present a self-contained combinatorial proof of the lower bound theorem for triangulated manifolds, including a treatment of the cases of equality in this theorem.