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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?

Bruno Stonek
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    If I recall correctly, $\Sigma^n X$ is (at least homotopy equivalent to) $S^n \wedge X$, so there is a natural bijection between $[\Sigma^n X, Y]$ and $[S^n, \langle X, Y \rangle]$, where $\langle X, Y \rangle$ is the space of pointed maps $X \to Y$ (when this exists). Thus they are homotopy groups. – Zhen Lin Feb 07 '14 at 16:16
  • @ZhenLin: yes, but of a different space! In any case, thanks, that's a nice observation. – Bruno Stonek Feb 07 '14 at 16:18
  • You are asking two very different questions... You should split them. Actually just keep the first question, because the second one has already been answered on this site, see http://en.wikipedia.org/wiki/Whitehead_theorem#Spaces_with_isomorphic_homotopy_groups_may_not_be_homotopy_equivalent and http://math.stackexchange.com/questions/88943/two-cw-complexes-with-isomorphic-homotopy-groups-and-homology-yet-not-homotopy and http://math.stackexchange.com/questions/56500/weak-homotopy-equivalence-whitehead-theorem-and-the-pseudocircle (for the non CW case) – Najib Idrissi Feb 07 '14 at 16:31
  • @nik: as interesting as those questions may be, I don't see any reference to the groups I allude to in this one. – Bruno Stonek Feb 07 '14 at 16:40
  • @nik: I understand what you're saying now. But the questions are the end are to be taken to mean: "in these circumstances, does the knowledge of these groups for all $X$ give us information?" (the questions you linked being taking $X=S^0$) – Bruno Stonek Feb 07 '14 at 17:03

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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 $X$.

Qiaochu Yuan
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