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.

Making a dunce hat, by Claudio Rocchini. From http://goo.gl/bCvuE

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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s