# Triangulations of exotic 4-spheres

Are there explicit examples of triangulations of exotic 4-spheres?

• I was unaware that exotic 4-spheres existed. That seems to be the smooth Poincaré conjecture. – S. Carnahan May 29 '10 at 8:14
• – Spinorbundle May 29 '10 at 8:25
• OK, thanks. And what about higher-dimensional exotic spheres? – John Vrem May 29 '10 at 15:03
• The section on "Explicit examples of exotic spheres" in the wikipedia article describes Milnor's 7-sphere. – j.c. May 29 '10 at 17:21
• I thought "exotic sphere" just meant a differentiable structure on the standard topological sphere which is inequivalent to the standard differentiable structure. So all you have to do is triangulate the standard topological sphere. Am I making a dumb mistake? – Todd Trimble May 29 '10 at 18:59

In general, explicit triangulations of higher dimensional manifolds seem to be difficult to write down. I've heard from computer algebra specialists that no one has even written an explicit triangulation of $\mathbb{CP}^3$. The chaos surrounding this earlier question might suggest that the problem is subtle.
• I somehow doubt that exhibiting vertices of an explicit triangulation of $\mathbb{CP}^3$ is hard. Of course, any statement of the form "no one has ever written blah" needs to be interpreted cautiously. – Victor Protsak May 29 '10 at 22:11
• Thank you for clarifying! So if I understand correctly, they are trying to $\textit{glue}$ a manifold from small pieces, whereas the most natural way in the case of $\mathbb{CP}^n$ would seem to be to $\textit{subdivide}$ it until one gets convex polyhedra (and there may well be other ways). Of course, the point about the vertex count is a valid one. – Victor Protsak May 30 '10 at 6:14