  We review the theory of combinatorial intersection cohomology of fans
developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This
theory gives a substitute for the intersection cohomology of toric varieties
which has all the expected formal properties but makes sense even for
non-rational fans, which do not define a toric variety. As a result, a number
of interesting results on the toric $g$ and $h$ polynomials have been extended
from rational polytopes to general polytopes. We present explicit complexes
computing the combinatorial IH in degrees one and two; the degree two complex
gives the rigidity complex previously used by Kalai to study $g_2$. We present
several new results which follow from these methods, as well as previously
unpublished proofs of Kalai that $g_k(P) = 0$ implies $g_k(P^*) = 0$ and
$g_{k+1}(P) = 0$.
