A characterization of simplicial polytopes with g₂=1

Kalai proved that the simplicial polytopes with g₂=0 are the stacked polytopes. We characterize the g₂=1 case.Specifically, we prove that every simplicial d-polytope (d≥4) which is prime and with g₂=1 is combinatorially equivalent either to a free sum of two simplices whose dimensions add up to d (each of dimension at least 2), or to a free sum of a polygon with a (d-2)-simplex. Thus, every simplicial d-polytope (d≥4) with g₂=1 is combinatorially equivalent to a polytope obtained by stacking over a polytope as above. Moreover, the above characterization holds for any homology (d-1)-sphere (d≥4) with g₂=1, and our proof takes advantage of working with this larger class of complexes.