The homotopy groups $\pi\_{n}(X)$ arise from considering equivalence classes of based maps from the $n$-sphere $S^{n}$ to the space $X$. As is well known, these maps can be composed, giving arise to a group operation. The resulting group contains a great deal of information about the given space. My question is: is there any extra information about a space that can be discovered by considering equivalence classes of based maps from the $n$-tori $T^{n}=S^{1}\times S^{1}\times \cdots \times S^{1}$. In the case of $T^{2}$, it would seem that since any path $S^{1}\to X$ can be "thickened" to create a path $T^{2}\to X$ if $X$ is three-dimensional, the group arising from based paths $T^{2}\to X$ would contain $\pi\_{1}(X)$. Perhaps more generally, can useful information be gained by examining equivalence classes of based maps from some arbitrary space $Y$ to a given space $X$.



A possible generalization of the homotopy groups.
There's always information to be got. But in this case:


* Based homotopy classes of maps $T^2\to X$ don't form a group! To define a natural function $\mu\colon [T,X]\_\*\times [T,X]\_\*\to [T,X]\_\*$, you need a map $c\colon T\to T\vee T$ (where $\vee$ is one point union). And if you want $\mu$ to be unital, associative, etc., you'll want $c$ to be counital, coassociative, etc. For $T=T^n$ with $n\geq2$, there is no $c$ that is counital. (The usual way to see this is to think about the cohomology $H^\*T$ with its cup-product structure.)
* The inclusion $S^1\vee S^1\to T^2$ gives a map
$$r\colon [T^2,X]\_\* \to [S^1\vee S^1,X]\_\*\approx \pi\_1X\times \pi\_1X.$$
The *image* of this map will be pairs $(a,b)$ of elements in $\pi\_1X$ which commute: $ab=ba$. It won't usually be injective; so there might be something interesting to think about the in preimages $r^{-1}(a,b)$.


