I need to prove the following $\pi_1\left(f_T(\mathbb{S}^2), f_T(\mathbf{x}_0)\right)\simeq \pi_1\left(\mathbb{S}^2, \mathbf{x}_0\right)$.
My idea is to prove that the function $f_T$ is an homeomorphism, i need prove $f_T$ is continuous and has an inverse.
Let the matrix $T$ $\in \mathbb{GL}_3(\mathbb{C})\cap\mathbb{R}^{3\times 3}$ such that $T\mathbf{x}\in \mathbb{S}^2$ for every $\mathbf{x}\in \mathbb{S}^2$. Then consider the function $f_T:\mathbb{S}^2 \to \mathbb{S}^2, y\to Ty$