Consider the unit 3-sphere given by coordinates (x,y,z,w). The subspace (x^2 + y^2 ≤ 1/2) is a solid torus, and we can project it to a disk, and then onto a 2-sphere. We map the rest of the 3-sphere to the basepoint of the 2-sphere.
Now this map is the Hopf fibration. And in my opinion the easiest way to see this is via the cup product structure in the James construction. But is there a way to directly compute the homotopy to the standard Hopf fibration?
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