If $X,Y$ are pointed spaces, denote by $[X,Y]$ the pointed homotopy classes of pointed maps $X\to Y$.
The sets $[\Sigma^nX,Y]$ actually have the structure of a group for $n\geq 1$. Here $\Sigma$ denotes reduced suspension.
If we take $X=S^0$, then these groups are isomorphic to $\pi_n(Y)$. So the preceding groups generalize the homotopy groups.
Are the groups $[\Sigma^nX,Y]$ (which are not homotopy groups) studied actively? How much new information do they give that homotopy groups do not?
A result which is somewhat related to the question is Whitehead's theorem. It somehow says that under certain hypotheses, we can deduce that a map is a homotopy equivalence knowing that it induces an isomorphism on homotopy groups, so for this purpose the information given by the homotopy groups suffice (given that we have a map between the nice spaces).
But what if we don't actually have a map between the spaces? What if they don't have the homotopy type of a CW complex?