  We recently defined a property of Morse shellability (and tileability) of
finite simplicial complexes which extends the classical one and its relations
with discrete Morse theory. We now prove that the product of two Morse tileable
or shellable simplicial complexes carries Morse tileable or shellable
triangulations under some tameness condition, and that any tiling or shelling
becomes tame after one barycentric subdivision. We deduce that any finite
product of closed manifolds of dimensions less than four carries Morse
shellable triangulations whose critical and h-vectors are palindromic. We also
prove that the h-vector of a Morse tiling is always palindromic in dimension
less than four or in the case of an h-tiling, provided its critical vector is
palindromic.
