TITLE: The groups $[\Sigma^nX,Y]$ versus the homotopy groups
QUESTION [9 upvotes]: 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?

REPLY [4 votes]: Yes, they are actively studied, because people care about the homotopy type of mapping spaces (which is what you are computing, as Zhen Lin says in the comments). For example, if you take $Y$ to be an Eilenberg-MacLane space then you are computing the cohomology of $X$.