# Triangulation of the complex projective plane

In the 1983 paper The 9-vertex Projective Plane'' by W. Kuehnel and T.F. Banchoff (The Mathematical Intelligencer Vol 5.) the authors give a 9 vertex triangulation of the complex projective plane, which they call $$\mathbb{CP}_9^2$$, whose details I won't give here. The crux of their argument is:

1) $$\mathbb{CP}^2$$ can be represented as the mapping cone of the Hopf map $$h:S^3\to S^2$$

2) Their simplicial complex has the duality property that every 4-simplex $$\sigma$$ spanned by 5 vertices is opposite to a non-bounding 2-cycle which is the boundary of the 3-simplex $$\rho$$ spanned by the vertices not in $$\sigma$$.

3) Any point in $$\mathbb{CP}_9^2\backslash \{\sigma\cup\partial \rho\}$$ can uniquely be written as $$tv_0+(1-t)v_1$$ for some $$v_0\in \sigma$$ and $$v_1\in \partial \rho$$ and $$t\in (0,1)$$.

4) The key point: the level sets defined by a fixed $$t$$ are actually simplicial 3-spheres which approach $$\sigma$$ as $$t\to 1$$ and as $$t\to 0$$ they approach a decomposition of $$S^3$$ into four solid tori which are mapped to the faces of $$\partial \rho$$.

My question is why are the level sets 3-spheres? From their paper, we can see that the pre-image of each face of $$\partial \rho$$ under the Hopf map from $$\sigma$$ to $$\partial \rho$$ is a 2-sphere up to homotopy equivalence so I guess the level sets are also homotopy equivalent to $$S^3$$. But why would they be homeomorphic to $$S^3$$? The authors do not give a complete proof, but instead say that the proof uses a collapsing'' argument. What collapsing argument would imply that something is homeomorphic to $$S^3$$?

The picture they give is the following

The numbers on the edges represent the faces of $$\partial \rho$$ and the edges are pre-images.

EDIT: In this paper, the authors give an parallel example for the $$\mathbb{RP}^2$$ triangulation. The following picture shows a well-known triangulation of $$\mathbb{RP}^2$$. There is a distinguished simplex 123 and the opposite non-bounding cycle given by the boundary of 456 (shown in green). A level set is given in blue. It is clear from the picture that the level sets are 1-spheres that approach a double cover of $$\partial(456)$$. But it much less clear in the case of $$\mathbb{CP}^2_9$$ that the level sets are 3-spheres! Also, just because one complex collapses to another doesn't mean that they are homeomorphic! $\mathbb{RP}^2$">

• How can $\partial\rho$ decompose into solid tori? It is the boundary of a 3-simplex, thus a 2-sphere. – ThiKu Dec 14 '18 at 23:10
• Can you elaborate what is the appearance of the Hopf map? – ThiKu Dec 14 '18 at 23:11
• @ ThiKu. I'm sorry, it actually approaches a decomposition of the 3-sphere, which are mapped to the four faces of $\partial \rho$. – Evan Wilson Dec 14 '18 at 23:21
• @ ThiKu. I think the Hopf map appears as the map which takes a point in $\sigma$ at $t=1$ to the corresponding point in $\partial \rho$ as $t\to 0$. – Evan Wilson Dec 14 '18 at 23:38
• Likely not relevant, but I wonder if the vertices are the 9 points of a "SIC-POVM"? aip.scitation.org/doi/10.1063/1.1737053 en.wikipedia.org/wiki/SIC-POVM – Ian Agol Dec 15 '18 at 5:32