  A connected sum construction for local rings was introduced in a paper by H.
Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein
case, this can be viewed as an algebraic analogue of the topological
construction of the same name. We give two alternative description of this
algebraic connected sum: the first uses algebraic analogues of Thom classes of
vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual
generators. We also investigate the extent to which the connected sum
construction preserves the weak or strong Lefschetz property, thus providing
new classes of rings which satisfy these properties.
