Geometrically, the Hopf map is a continuous map from $S^3$ onto $S^2$, with fibres $S^1$. It can also be viewed as linking complex with real geometry, in some sense. Indeed, $S^3$ can be thought of sitting inside $\mathbb{C}^2$, and the domain of the Hopf map can easily be extended to give a continuous map from $\mathbb{C}^2 \setminus \{0\}$ onto $S^2$, with $\mathbb{C}^*$ fibres.

Let us say that instead of $\mathbb{R}$, we had some field $k$, which could for instance be finite. Is there an analogue of the Hopf map for such a field $k$, under perhaps some conditions? One way to make it more precise is this. Is there a field extension $L$ of $k$, such that there exists (under some conditions) a morphism from the affine 2-space over $L$ minus the origin onto the 2-sphere over $k$, and sharing as many properties as possible with the classical Hopf map?

  • $\begingroup$ In $\mathbb{A}^1$-homotopy theory, the Hopf map is almost what you said : replace $S^3$ by $k^2\setminus\{0\}$, and replace $S^2$ by $\mathbb{P}^1_k$. The Hopf map is then the canonical projection with $k^*$ fibres. Note that $\mathbb{P}^1$ is topologically $S^2$ when $k=\mathbb{C}$. $\endgroup$ – Roland Jun 13 '17 at 14:56
  • $\begingroup$ @Roland yes thank you. I knew that, but I am interested in the case where the target space can be identified as a 2-sphere over some other field (say a subfield). $\endgroup$ – Malkoun Jun 13 '17 at 14:59
  • $\begingroup$ What do you mean by 2-sphere ? $\operatorname{Spec} k[x,y,z]/(x^2+y^2+z^2-1)$ ? (The thing is, this scheme does not share a lot of properties with the topological $S^2$) $\endgroup$ – Roland Jun 13 '17 at 15:03
  • $\begingroup$ Let me think a little which properties I am interested in having. I am interested in having as target space the space of "rays" starting from the origin in some affine 3-space over a field. But I am not sure how to formalize the notion of rays for finite fields. $\endgroup$ – Malkoun Jun 13 '17 at 15:09
  • $\begingroup$ If instead of rays you would be happy with lines thru the origin, then the affine 3-space (sans origin) moded out by the equivalence relation of "being on the same line" gives you simply the projective plane over $k$. OTOH in the Hopf map we really have, IIRC, the 4D-algebra of quaternions moded out by half the center, giving $S^3$, further moded out by the maximal commutative subfield $\Bbb{C}^*$. This fails on the finite field side, because all f.d. division algebras are commutative. Hmm. Is that really an obstacle? Hmmm... $\endgroup$ – Jyrki Lahtonen Jun 13 '17 at 15:26

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