Is the inclusion of the maximal torus in a simply connected compact Lie group null-homotopic?

Question

Let $G$ be a simply connected compact Lie group and $T$ its maximal torus with inclusion $i:T \hookrightarrow G$.

By simply connectedness of the group $G$ and asphericity of the torus $T$, the induced map $i_* : \pi_*(T) \to \pi_*(G)$ is trivial. Moreover, the maximal torus of $S^3$ or of $Spin(4) \cong S^3 \times S^3$ has a null-homotopic inclusion into the respective group.

Therefore, I am wondering if the above indicates that this is true in greater generality, i.e. is the inclusion $i:T\hookrightarrow G$ of the maximal torus in a simply-connected Lie group null-homotopic?

Answer 1

Yes.

Write $T=T_1\times\dots\times T_k$, with each $T_i$ isomorphic to the circle group. Since $T_i$ is a circle and $G$ is simply connected, there exists a homotopy $(f_i(t,x))$, with $f_i(0,x)=x$ and $f_i(1,x)=1$ for $x\in T_i$.

Then for $x=(x_1,\dots,x_k)\in T$, write $f(t,x)=\prod_{i=1}^kf_i(t,x)$. Then $f(0,x)=x$ and $f(1,x)=1$.