The standard -simplex is the set . Let’s call two vectors and equivalent if they become equal after all zero entries are deleted. For example, . The axioms of the equivalence relation are obviously satisfied. The quotient space is a compact topological space. How does it look?
When , the simplex is a line segment, and the equivalence relation glues its endpoints together. Thus, is the circle.
When , the simplex is a triangle. Gluing two sides that share the vertex creates a cone, such as on the cover of Fastball debut album. The remaining identification takes an effort to carry out: (1) bend the cone so that the vertex touches the base, making the previous glue line into a circle. (2) glue this circle to the base of the cone. The result is a space in which all three former edges become a single circle; clearly this is not a manifold. It is the topological dunce hat. Despite the difficulty of twisting and gluing, one can realize this space as a subset of .
When , I don’t know if is homeomorphic to a subset of . Actually, one should ask for bi-Lipschitz equivalence, since carries a natural quotient metric. The question stems from a 1931 paper by Borsuk and Ulam. The term “higher dimensional dunce hat” was introduced by Andersen, Marjanović and Schori, although they apply it only to even-dimensional hats. They prove that is contractible when is even, and has the homotopy type of when is odd. (This does not imply that is homeomorphic to : the Poincaré conjecture concerns manifolds, and is not a manifold when ).
Zeeman conjectured that the product of any contractible 2-complex (such as ) with is collapsible. This statement implies the Poincaré conjecture, but … oh well, so much for this chance to make your mama proud.