  In this paper we define a family of topological spaces, which contains and
vastly generalizes the higher-dimensional Dunce hats. Our definition is purely
combinatorial, and is phrased in terms of identifications of boundary simplices
of a~standard d-simplex. By virtue of the construction, the obtained spaces may
be indexed by words, and they automatically carry the structure of a
$\Delta$-complex.
  As our main result, we completely determine the homotopy type of these
spaces. In fact, somewhat surprisingly, we are able to prove that each of them
is either contractible or homotopy equivalent to an odd-dimensional sphere. We
develop the language to determine the homotopy type directly from the
combinatorics of the indexing word.
  As added benefit of our investigation, we are able to emulate the Dunce hat
phenomenon, and to obtain a large family of both $\Delta$-complexes, as well as
simplicial complexes, which are contractible, but not collapsible.
