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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.

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4 Answers 4

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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.

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Actually, with simplicial sets, you can probably find a significantly smaller model, since simplices don't have to be determined by their vertices and can have degenerate boundaries. –  Eric Wofsey Oct 28 '09 at 17:52
This seems possible. –  Benjamin Antieau Oct 28 '09 at 19:54
You may also check "A nontrivial pairing of finite T_0 spaces" by Hardie,Vermeulen and Witbooi and "Crown multiplications and a higher order Hopf construction" by Hardie and Witbooi. –  Michał Kukieła Apr 19 '11 at 9:55

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?)

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I think you're thinking of his paper "A combinatorial definition of homotopy groups" from 1958, which (in section IV.21) computes pi_3(S^2) using his simplicial-group techniques. –  Tyler Lawson Oct 28 '09 at 16:18

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.

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I have a question about this construction. The simplicial set Y will be infinite, and not a Kan complex. However, one might hope that there is a finite simplicial set K and a map f: K-> Y, so that the geometric realization of K is S^3, and so that f is a weak equivalence. Then K->S^2 would be a finite model of the Hopf map. How hard is it to find such K and f? –  Charles Rezk Oct 28 '09 at 16:57
It should follow from classical simplicial approximation that such a finite K and f do exist, though that doesn't help you find them. –  Eric Wofsey Oct 28 '09 at 17:15

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.

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