How much is cohomotopy dual to homotopy? To what degree can we dualize theorems regarding homotopy into theorems about cohomotopy (or is there a good source that tries to do this)?
For instance, is there some kind of Hurewicz theorem relating cohomotopy and ordinary cohomology? Is there a "cohomotopy extension property" (something that applies when relative cohomotopy groups are trivial)? If two spaces are cohomologically equivalent and have some property in cohomotopy analogous to simply-connected, are they cohomotopy equivalent?
Thanks, this is primarily a reference request, however there is the possibility that all this is impossible so no such reference exists, which would also be an acceptable answer. 

The homotopy groups can be written as covariant homotopy invariant functors $\pi_n:\mathrm{Top}_\ast\to\mathrm{Set}$. If we were to consider contravariant homotopy invariant functors $\pi^n:\mathrm{Top}_\ast^{op}\to\mathrm{Set}$, we would obtain the cohomotopy sets. How dual is it? Well, $\pi^n(S^m)=\pi_m(S^n)$. If $X$ is a CW-complex of dimension (at most) $n$, then $\pi^p(X)\to H^p(X)$ is a bijection. See the nlab. As for whether or not a "cohomotopy extension property" exists, I don't know; it seems like an interesting thing!
