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I understand the duality in the case of homology and de Rham cohomology. Through integration a chain can be understood as a linear functional on differential forms and vice versa.

Is there a way to understand cohomotopy classes as functionals on a homotopy group? Is there a different sense in which it is dual?

I am aware of a similarly titled question but that was more about relating cohomotopy to cohomology rather than to homotopy.

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    $\begingroup$ The cohomotopy set $\pi^n(X)$ is in general not even a group for $n \ge 2$, so it's going to be a weird kind of duality if it exists... Are you talking about stable cohomotopy? $\endgroup$ Commented Feb 13, 2015 at 19:12
  • $\begingroup$ Are you looking for some kind of deeper duality than the fact that the homotopy groups are defined by $\pi_n(X)=[S^n,X]$ and the cohomotopy by $\pi^n(X)=[X,S^n]$? $\endgroup$
    – Dan Rust
    Commented Feb 16, 2015 at 14:39
  • $\begingroup$ @DanielRust, Yes (unless I'm misunderstanding your notation) I was hoping there was something deeper behind the name. Like a way to say something about cohomotopy through the structure of $\pi_n$ itself, rather than having to refer back to $X$, or something of that sort. $\endgroup$
    – octonion
    Commented Feb 16, 2015 at 21:36

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