A manifold which is not homeomorphic to a sphere but
has the same homology group as a sphere.
This is derived by suitable identifications of facets
of a regular dodecahedron.
(Identify the vertices with the same label in the figure below.)
The data given here is a triangulation made by
Björner and Lutz.
This is the smallest triangulation currently known.
(16 vertices and 90 facets)
This is Cohen-Macaulay, but not constructible nor shellable.