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Back in the 1940's, Ralph Fox defined something called the torus homotopy group. For a based space $(Y,y_0)$ and natural number $r$, the $r$-dimensional torus homotopy group $\tau_r(Y,t_0)$ \tau_r(Y,y_0)$is just the fundamental group of the mapping space${\rm map}(T^{r-1},Y)$, based at the constant map (where$T^{r-1}$is of course a torus). The group$\tau_r(Y,t_0)$\tau_r(Y,y_0)$ contains isomorphic copies of $\pi_n(Y,y_0)$ for all $n\leq r$. Also, Whitehead products become commutators in the torus homotopy group. By passing to the limit over $r$ one obtains the (infinite) torus homotopy group $\tau(Y, y_0)$, which contains all of the homotopy information of $Y$ in one place!

Unfortunately for Fox, the idea doesn't seem to have caught on (although I hear he had a few others which did). MathSciNet only turns up 11 papers containing the phrase "torus homotopy groups" (although the most recent is from 2007).

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Back in the 1940's, Ralph Fox defined something called the torus homotopy group. For a based space $(Y,y_0)$ and natural number $r$, the $r$-dimensional torus homotopy group $\tau_r(Y,t_0)$ is just the fundamental group of the mapping space ${\rm map}(T^{r-1},Y)$, based at the constant map (where $T^{r-1}$ is of course a torus).

The group $\tau_r(Y,t_0)$ contains isomorphic copies of $\pi_n(Y,y_0)$ for all $n\leq r$. Also, Whitehead products become commutators in the torus homotopy group. By passing to the limit over $r$ one obtains the (infinite) torus homotopy group $\tau(Y, y_0)$, which contains all of the homotopy information of $Y$ in one place!

Unfortunately for Fox, the idea doesn't seem to have caught on (although I hear he had a few others which did). MathSciNet only turns up 11 papers containing the phrase "torus homotopy groups" (although the most recent is from 2007).