# Dunce hats in assorted dimensions

The standard $n$-simplex is the set $\Delta^n=\{x\in\mathbb R^{n+1}\colon \forall i\ x_i\ge 0,\ \sum_{i=1}^{n+1} x_i=1\}$. Let’s call two vectors $(x_1,\dots,x_{n+1})$ and $(x_1',\dots,x_{n+1}')$ equivalent if they become equal after all zero entries are deleted. For example, $(1/3,0,2/3,0,)\sim (0,1/3,2/3,0)$. The axioms of the equivalence relation are obviously satisfied. The quotient space $\Delta^n/\sim$ is a compact topological space. How does it look?

When $n=1$, the simplex is a line segment, and the equivalence relation glues its endpoints together. Thus, $\Delta^1/\sim$ is the circle.

When $n=2$, the simplex is a triangle. Gluing two sides that share the vertex $(1,0,0)$ 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 $\mathbb R^3$.

When $n\ge 3$, I don’t know if $\Delta^n/\sim$ is homeomorphic to a subset of $\mathbb R^{n+1}$. Actually, one should ask for bi-Lipschitz equivalence, since $\Delta^n/\sim$ 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 $\Delta^n/\sim$ is contractible when $n$ is even, and has the homotopy type of $S^{n}$ when $n$ is odd. (This does not imply that $\Delta^3/\sim$ is homeomorphic to $S^3$: the Poincaré conjecture concerns manifolds, and $\Delta^n/\sim$ is not a manifold when $n\ge 2$).
Zeeman conjectured that the product of any contractible 2-complex (such as $\Delta^2/\sim$) with $[0,1]$ is collapsible. This statement implies the Poincaré conjecture, but … oh well, so much for this chance to make your mama proud.