On homology spheres with few minimal non-faces

Abstract

Let Δ \Delta be a ( d − 1 ) (d-1) -dimensional homology sphere on n n vertices with m m minimal non-faces. We consider the invariant α ( Δ ) = m − ( n − d ) \alpha (\Delta ) = m - (n-d) and prove that for a given value of α \alpha , there are only finitely many homology spheres that cannot be obtained through one-point suspension and suspension from another. Moreover, we describe all homology spheres with α ( Δ ) \alpha (\Delta ) up to four and, as a corollary, all homology spheres with up to eight minimal non-faces. To prove these results we consider the lcm-lattice and the nerve of the minimal non-faces of Δ \Delta . Also, we give a short classification of all homology spheres with n − d ≤ 3 n-d \leq 3 .