A characterization of simplicial polytopes with

Kalai proved that the simplicial polytopes with g 2 = 0 are the stacked polytopes. We characterize the g 2 = 1 case. ically, 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.