The action of the orthogonal group on the sphere is not formal

Published .
math

In this post, I record a simple and probably well-known fact; but since I have to remake the computation again and again (because I forget it…) I thought it would be nice to have it in an accessible place.

The fact is that for an odd n3n \ge 3, the usual action of the special orthogonal group SO(n)SO(n) on the sphere Sn1S^{n-1} is not formal over Q\mathbb{Q} in the sense of rational homotopy theory, even though both spaces are formal. This was first told to me by Thomas Willwacher, and it is mentioned as Remark 9.5 in his paper “Real models for the framed little (n)-disks operads” (arXiv:1705.08108) with Anton Khoroshkin.

Here is a quick proof for n=3n=3 that generalizes easily. Suppose that the action map ρ:SO(n)×Sn1Sn1\rho : SO(n) \times S^{n-1} \to S^{n-1} is formal. Then one can find maps that make the following diagram commute:

Ω(S2)ρΩ(SO(3)×S2)Ω(SO(3))Ω(S2) H(S2)ρH(SO(3))H(S2)=H(SO(3))H(S2)\begin{CD} \Omega(S^2) @>{\rho^*}>> \Omega(SO(3) \times S^2) @<<< \Omega(SO(3)) \otimes \Omega(S^2) \\ @A{\sim}AA @. @A{\sim}AA \\ H(S^2) @>{\rho^*}>> H(SO(3)) \otimes H(S^2) @= H(SO(3)) \otimes H(S^2) \end{CD}

Let υH2(S2)\upsilon \in H^2(S^2) be a generator. By degree reasons, we must have ρ(υ)=λυ\rho^*(\upsilon) = \lambda \otimes \upsilon for some constant λQ\lambda \in \mathbb{Q}. This implies that, over Q\mathbb{Q}, the map ρ\rho factors through the second projection p2:SO(3)×S2S2p_2 : SO(3) \times S^2 \to S^2.

We know that pSO(3)QS3pSO(3) \simeq_{\mathbb{Q}} S^3 and that π3Q(S3)=Q\pi^{\mathbb{Q}}_3(S^3) = \mathbb{Q}. Let xx be a generator and let us show that ρ(x×1)\rho_(x \times 1) must generate πQ3(S2)\pi^{\mathbb{Q}}*3(S^2) We also know that ρ\rho induces a principal SO(2)SO(2)-bundle :

SO(2)SO(3)S2SO(2) \hookrightarrow SO(3) \twoheadrightarrow S^2

where p:SO(3)S2p : SO(3) \to S^2 is given by p(A)=Awp(A) = A \cdot w for a fixed vector ww and SO(2)SO(2) is the stabilizer of ww. We thus get a long exact sequence of homotopy groups:

π3(SO(2))π3(SO(3))pπ3(S2)π2(SO(2)).\pi_{3}(SO(2)) \to \pi_{3}(SO(3)) \xrightarrow{p_*} \pi_{3}(S^{2}) \to \pi_{2}(SO(2)).

Since π3(SO(2))=π2(SO(2))=0\pi_{3}(SO(2)) = \pi_{2}(SO(2)) = 0, we get that p(x)=ρ(x×1)p_*(x) = \rho_*(x \times 1) is a generator. This contradicts the fact that ρ\rho factors through the second projection as then ρ(x×1)\rho_*(x \times 1) would vanish.