We know that the homotopy class of the maps $S_d \to X$ is $\pi_d(X)$. What is the homotopy class of the maps $(S_1)^d \to X$? Here $X$ can be an $n$dimensional sphere $S_n$, or a classifying space $BG$ of some simple groups $G=Z_2,U(1),SU(n)$ etc.
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