  Kalai proved that the simplicial polytopes with g_2=0 are the stacked
polytopes. We characterize the g_2=1 case. Specifically, we prove that every
simplicial d-polytope (d>=4) which is prime and with g_2=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 g_2=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 g_2=1, and our proof takes advantage of working with
this larger class of complexes.
