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.

The cases of $BG$ where $G=\mathbb{Z}/2$ and $U(1)$ are easy, since these are $K(\mathbb{Z}/2,1)$ and $K(\mathbb{Z},2)$ respectively, so you get $H^1(T^d,\mathbb{Z}/2)$ and $H^2(T^d,\mathbb{Z})$ (when looking at pointed homotopy classes).
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Mark GrantFeb 8 '13 at 7:09

2

For $d=2$, there is an action of $\pi_2(X)$ on $[S^1\times S^1,X]$ whose quotient set is in bijection with $\{a,b\in\pi_1(X);a+b=b+a\}\subset[S^1\vee S^1,X]$ via the restriction along $S^1\vee S^1\subset S^1\times S^1$.
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Fernando MuroFeb 8 '13 at 7:42

1

@Dylan, tori split stably as wedges of spheres.
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Fernando MuroFeb 8 '13 at 8:56

2

@Xiao-Gang Wen: A general method for computing torus homotopy groups is given in Section 3.1 of the recent preprint arxiv.org/abs/1302.1826 by Lupton and Smith. It's worth pointing out though, that $\tau_d(X)$ is not the same thing as $[T^d,X]$, which is what you originally asked about.
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Mark GrantFeb 22 '13 at 17:03