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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.

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Usually symmetry issues are not discussed in discussions of being shellable but you may find the information on this web page, and the references given there of use: eg-models.de/models/Simplicial_Manifolds/2003.05.004/… –  Joseph Malkevitch Mar 31 '10 at 17:46
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