Simplicial model of Hopf map?

Question

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 $\mathbb{C}P^2$.

Answer 1

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.

Answer 2

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?)

Answer 3

There is a small simplicial set description of the Hopf map in Clemens Berger's thesis, Exemple 1.19, pp. 45-47.

Answer 4

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.

Answer 5

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')]

Answer 6

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.