Regular Representatives of Homotopy Classes Between Spheres

Definitions and mathematical statement

For each $r\in{\mathbb{Z}}_{\geq0}$, equip \[ S^r=\{x\in{\mathbb{R}}^{r+1}:x_0^2+\cdots+x_r^2=1\} \] with its standard real algebraic structure and Euclidean topology. In particular $S^0$ is the two-point sphere, and the integers $n,p$ below include zero.

The source conjecture asks whether, for every continuous map $f:S^n\to S^p$, there exist a regular map $g:S^n\to S^p$ and a continuous homotopy \[ H:S^n\times[0,1]\longrightarrow S^p, \qquad H(x,0)=f(x),\quad H(x,1)=g(x). \] The quantifiers are $\forall n,p\geq0\ \forall f\ \exists g,H$; the representative can depend on the map and dimension pair. No basepoint condition or regularity of the homotopy itself is required by the wording.

The term regular needs its real-algebraic convention. In a common real-algebraic category, it means coordinate functions locally given by rational functions with denominators nonzero on the relevant real neighborhoods. A polynomial restriction is regular, but an assertion specifically about polynomial representatives or morphisms of the full affine schemes can be stronger than this no-real-poles interpretation. The record does not define the convention, so these categories are not silently identified. The standard quadratic models above are likewise made explicit; replacing a sphere by a different algebraic model would be a different formulation without further comparison.

The task is existence of at least one regular representative in each continuous homotopy class. It is not a requirement that every continuous map already be regular, that regular maps be dense in all continuous maps, or that arbitrary regular representatives admit regular homotopies. Both $n$ and $p$ range over all nonnegative integers, without a dimension inequality.

The literature triage checked 2026-08-21 reports known results for selected dimension pairs and says the all-pairs assertion remains unresolved. It supplies no list or precise hypotheses of those cases. The open label and reported official-source verification remain dated metadata, not independently checked current conclusions.

Short English statement

Does every continuous homotopy class from any standard sphere to any other contain a regular real-algebraic map? The homotopy is only required to be continuous, and zero-dimensional spheres remain included. The source does not define whether regular uses the no-real-poles convention or the stronger polynomial/scheme setting.

Sources

• Local UnsolvedMath registry, permanent ID 30000713, source code catalogue entry, "Regular Representatives of Homotopy Classes Between Spheres," including the original conjecture, all nonnegative dimension pairs and verification metadata.
• Reelle Algebraische Geometrie (2007), Oberwolfach Reports, pp. 753--820, DOI 10.4171/owr/2007/14: https://doi.org/10.4171/owr/2007/14.
• Supplied reference "regular-map homotopy results": https://doi.org/10.1007/s00208-006-0061-3.
• Supplied reference "later sphere-map advances": https://doi.org/10.1016/j.matpur.2022.03.002.
• Literature triage checked 2026-08-21; record updated 2026-09-16. No external source has been visited or independently checked here.