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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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octonion
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    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? – Najib Idrissi Feb 13 '15 at 19:12
  • 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 $\pi^n(X)=[X,S^n]$? – Dan Rust Feb 16 '15 at 14:39
  • @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. – octonion Feb 16 '15 at 21:36

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