Abstract: Connected sums were defined for local Gorenstein algebras by Ananthnarayan Avramov-Moore (A-A-M) in a 2012 paper. In the graded Artinian case, this construction is related to a topological construction that pastes two manifolds together along a common submanifold. In this case, the A-A-M construction can be described using algebraic versions of the Thom class of the normal bundle of a submanifold. We discuss this description here, as well as an alternative description of the A-A-M construction using Macaulay duality.

     Using this Macaulay dual description, in conjunction with J. Watanabe's theory of higher Hessians, we reproduce a well known proof that standard graded connected sums over a field always preserve the strong Lefschetz property (SLP). While our examples show that general connected sums do not always preserve SLP, we conjecture that the standard graded ones always do. This is based on joint work with A. Iarrobino and A. Seceleanu (U. of Nebraska).