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 bouquets of $n$-circles? 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 ...

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 ...