Simplicial model of Hopf map? 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 $\mathbb{C}P^2$.
 A: Here is one thing to try.  Start with the smallest simplicial model for S1 (the 1-simplex modulo its boundary).  Take the free group in each degree (but force the basepoint to be the identity).  The resulting simplicial group FS1 is a model for ΩS2; furthermore, being a simplicial group, it's a Kan complex.  Thus, we know there must be some map f: S2->FS1 which represents the generator of π2ΩS2; the group of FS1 in degree 2 is not too big, so it should not be hard to write this down explicitly (I haven't tried, though.)
Of course, you really want a map S3->X, where X models the 2-sphere.  Since FS1 is a simplicial group, let X=BF1, it's classifying space.  X is a model for the 2-sphere, and I expect that if you examine it closely, you will see the "suspension" of f corresponds to some explicit 3-simplex in X, which is your model.
I'm not sure this counts as a "combinatorial model", of course.
(I have a vague memory that Dan Kan did something like this in one of his papers in the 50s.  Is that right?)
A: There is a small simplicial set description of the Hopf map in Clemens Berger's thesis, Exemple 1.19, pp. 45-47.
A: Here's a different answer.  The Hopf fibration S3 -> S2 is a principal U(1)-bundle, which means it is the pullback of the universal U(1)-bundle along a map S2->BU(1).  
There is a simplicial model E->B of the universal fibration over BU(1) which is a Kan fibration:  since BU(1) is K(Z,2), you can take B to be a simplicial abelian group associated to the chain complex C concentrated in degree 2, and E is the simplicial abelian group associated to an acyclic complex A which has a surjective map to C.  Now pull back along S2->B and get a bundle Y->S2, and there you are.  The simplicial set Y will be a model for S3.
A: You ought to be able to trivialize the bundle over each hemisphere and loop at the transition function on the equatorial S^1 (which is presumably the identity map S^1 \to S^1 acting as rotations on the fiber).  Using this, it shouldn't be hard to write down an explicit geometric simplicial approximation to the Hopf map.  Alternatively, you could model S^1 as a simplicial group (the free abelian group on a 1-simplex and its degeneracies) and get a simplicial principal bundle on S^2 (which you should be able to model with one 0-simplex, one 1-simplex (the equator), and two 2-simplices) from this transition function.
A: As Benjamin Antieau pointed out, there is an explicit triangulation of $\mathbb{S}^3$ and $\mathbb{S}^2$ and the Hopf map as a simplicial map between those described in a paper by Madahar and Arkaria called A minimal triangulation of the Hopf map and its application. The paper describes how to construct the simplicial complexes, but does not list the simplices explicitly.
For the convenience of anyone looking for it, below I give a list of the tetrahedra of their triangulation of $\mathbb{S}^3$ (which describes it fully). The triangulation of $\mathbb{S}^2$ is just the boundary of the tetrahedron abcd, and the Hopf map is given on vertices by
$$
a_i\mapsto a,\ \ \  b_i\mapsto b,\ \ \  c_i\mapsto c,\ \ \  d_i\mapsto d
$$
for all $i\in\{0,1,2\}$.
[('a0', 'a1', 'b1', 'c1'),
 ('a0', 'a1', 'b1', 'd2'),
 ('a0', 'a1', 'c1', 'd2'),
 ('a0', 'a2', 'b0', 'c2'),
 ('a0', 'a2', 'b0', 'd1'),
 ('a0', 'a2', 'c2', 'd1'),
 ('a0', 'b0', 'b1', 'c1'),
 ('a0', 'b0', 'b1', 'd1'),
 ('a0', 'b0', 'c0', 'c1'),
 ('a0', 'b0', 'c0', 'c2'),
 ('a0', 'b1', 'd0', 'd1'),
 ('a0', 'b1', 'd0', 'd2'),
 ('a0', 'c0', 'c1', 'd2'),
 ('a0', 'c0', 'c2', 'd2'),
 ('a0', 'c2', 'd0', 'd1'),
 ('a0', 'c2', 'd0', 'd2'),
 ('a1', 'a2', 'b1', 'c1'),
 ('a1', 'a2', 'b1', 'd2'),
 ('a1', 'a2', 'c1', 'd2'),
 ('a2', 'b0', 'b2', 'c2'),
 ('a2', 'b0', 'b2', 'd2'),
 ('a2', 'b0', 'd1', 'd2'),
 ('a2', 'b1', 'b2', 'c2'),
 ('a2', 'b1', 'b2', 'd2'),
 ('a2', 'b1', 'c1', 'c2'),
 ('a2', 'c1', 'c2', 'd1'),
 ('a2', 'c1', 'd1', 'd2'),
 ('b0', 'b1', 'c1', 'd1'),
 ('b0', 'b2', 'c2', 'd2'),
 ('b0', 'c0', 'c1', 'd2'),
 ('b0', 'c0', 'c2', 'd2'),
 ('b0', 'c1', 'd1', 'd2'),
 ('b1', 'b2', 'c2', 'd2'),
 ('b1', 'c1', 'c2', 'd1'),
 ('b1', 'c2', 'd0', 'd1'),
 ('b1', 'c2', 'd0', 'd2')]

A: There is a paper [MathSciNet] 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.
