The idea of using maps from a sequence of simple standard objects into a topological space $X$ as $probes$ to explore its topology is ubiquitous. One organizes these maps into equivalence classes in such a way that the collection of classes acquires a nice algebraic structure. These algebraic invariants then serve to recognize $X$ or distinguish it from others.

One such sequence is, of course, pointed $n$-spheres, homotopy classes of maps from which yield homotopy groups, $\pi_n (X)$.

Has it been useful to consider other sequences of simple spaces for construction of invariants, e.g., homotopy classes of maps from $n$-tori, or from genus $n$ tori? Or can these always be simply expressed in terms of homotopy groups, and are, therefore, redundant? Or too hard to compute? Or lack good properties? Or ...