The Hopf fibration is a famous map S³ --> S² with fiber S¹, which is the generator in pi_3(S²). We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces and the induced map of simplcial 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 CP² out of S², so I would equivalently be interested in a small combinatorial model for CP².

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 $\mathbb{C}P^2$.