Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a nontrivial (non-constant) continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$? If yes, is there a map with this property which is not a null homotopic map?
2026-02-22 23:41:46.1771803706
Existence of a certain equivariant map from the sphere to a compact Lie group of lower dimension.
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What about$$\begin{array}{rccc}f\colon&S^2&\longrightarrow&S^1\\&(x,y,z)&\mapsto&\bigl(\cos(\pi x),\sin(\pi x)\bigr)?\end{array}$$