M. Rudin gave a triangulation of a tetrahedron with the property that after any small tetrahedron removed, the remaining part is not homeomorphic to a ball (An unshellable triangulation of a tetrahedron, Bull. Am. Math. Soc. (64), 1958, pp.~90--91). This example is not compatible with the tetrahedral rotation group $T_{12}$. Is there an unshellable triangulation invariant under the rotation group's action?

Some information (maybe usefull): P. Alfeld told me that RH Bing had another unshellable triangulation example using knot theory (Some aspects of the topology of 3-manifolds related to the Poincare conjecture, in "Lectures on Modern Mathematics", vol. 2, pp. 93-128, Wiley, New York, 1964). Bing's example is to prove that such triangulation does exist, but not a plicit construction.