The Hopf fibration is a famous map $S^3\to S^2$ with fiber $S^1$, which is the generator in $\pi\_3(S^2)$. We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces and the induced map of simplicial sets. But this model is **huge** and isn't really useful for doing calculations. Does anyone know a nice small model for this map in terms of simplicial sets? Something suitable for computations? This map is also the attaching map used to build $\mathbb{C}P^2$ out of $S^2$, so I would equivalently be interested in a small combinatorial model for $\mathbb{C}P^2$.



Simplicial model of Hopf map?
There is a paper [[MathSciNet](https://mathscinet.ams.org/mathscinet-getitem?mr=1789059 "Geom. Dedicata 82 (2000), no. 1-3, 105–114. zbMATH review at https://zbmath.org/0965.57021")] of Madahar and Arkaria called *A minimal triangulation of the Hopf map and its application*. They find a triangulation from a 12-vertex 3-sphere to a 4-vertex 2-sphere. The minimality is in Section 6.a. I hope this is useful.


Now, this gives the map the structure of a map of simplicial complexes. Choose an ordering of the vertices such that the map in the paper respects the order. This then gives you a model of the map on finite simplicial sets.


