Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$, such that $f$ is not a null homotopic map? This question was included in this MSE post but I did not receive any answer.
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$\begingroup$ Does "equivariant" in the title refer to the condition $f(-x)=f(x)^{-1}$? $\endgroup$– YCorApr 23, 2019 at 7:55
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The Blakers-Massey element in $\pi_6(S^3)\cong\mathbb{Z}_{12}$ can be represented by such a map. This is done explicitly on page 3 of the paper https://arxiv.org/abs/math/0501091, published as
Abresch, U.; Durán, C.; Püttmann, T.; Rigas, A., Wiedersehen metrics and exotic involutions of Euclidean spheres, J. Reine Angew. Math. 605, 1-21 (2007). ZBL1125.57017.
Let $\mathbb{H}$ denote the quaternions, and represent the $6$-sphere as $$ S^6 = \{(p,w)\in \mathbb{H}\times\mathbb{H} \mid \mathfrak{Re}(p)=0\mbox{ and } |p|^2+|w|^2=1\}. $$ The map $b:S^6\to S^3\subseteq \mathbb{H}$ is given by $$ b(p,w) = \left\{\begin{array}{ll} \frac{w}{|w|} e^{\pi p} \frac{\overline w}{|w|}, & w\neq 0 \\ -1, & w=0, \end{array}\right. $$ where $e^{\pi p} = \cos(\pi |p|) + \sin(\pi|p|) \dfrac{p}{|p|}$ is the quaternionic exponential.
The fact that $b(-p,-w)=\overline{b(p,w)}$ is easily checked (and is noted in the proof of Theorem 1 in the linked paper).
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3$\begingroup$ That's a very nice paper that I hadn't seen before, thanks for the pointer. $\endgroup$ May 2, 2018 at 9:28
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$\begingroup$ @Mark Is it realy $S^6$ ? Note that we are assuming $p\neq 0$ and $w \neq 0$. $\endgroup$ May 3, 2018 at 7:41
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2$\begingroup$ @AliTaghavi: It is. Both $p$ or $w$ can be $0$ (but not simultaneously). I was a bit sloppy before with the definition of $b$, I hope the edit clears the confusion. $\endgroup$ May 3, 2018 at 8:42