Recall that the cohomotopy set $\pi^k(\mathcal{M})$ is $[\mathcal{M},S^k]$, i.e., the set of pointed homotopy classes of continuous mappings $\mathcal{M}\to S^k$. Recall also the Whitehead theorem:



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> **Theorem:** *Suppose $X,Y$ are connected CW complexes. Suppose then that $f:X\to Y$ is a continuous map which induces an isomorphism $f\_{\*}:\pi\_k(X,x\_0)\to\pi\_K(Y,f(x\_0))$ for any $x\_0\in X$. Then $f$ is a homotopy equivalence.*
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**Does the following conjecture hold?**



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> **Conjecture:** *Suppose $X,Y$ are connected CW complexes. Suppose then that $f:X\to Y$ is a continuous map which induces an isomorphism $f^{\*}:\pi^k(Y)\to\pi^k(X)$. Then $f$ is a homotopy equivalence.*
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An obvious motivation comes from the Whitehead theorem. The only lead I have on finding a proof of this conjecture is a theorem of Hopf states that $\pi^k(X)$ is in bijection with $H^k(M)$.



Whitehead theorem for cohomotopy
No, this is false. According to the Sullivan Conjecture (Miller's Theorem), 
$\mathrm{map}\_\*(B\mathbb{Z}/p, S^n) \sim \*$ for all $n$, which means
$$
[\Sigma^n B\mathbb{Z}/p, S^k] = \*
$$
for all $n$. So if we let $f: \Sigma^k \mathbb{Z}/ p \to \*$, the induced map
$$
f^\*: 
\pi^k(\*) 
\to 
\pi^k ( \Sigma^n B\mathbb{Z}/p ) 
$$
is the equivalence $\* \to \*$. Since $f$ is not a homotopy equivalence, this counterexamps the conjecture.


Perhaps it would be more interesting to restrict attention to
maps $f:X\to Y$ between finite complexes.


EDIT (further thoughts): If $K$ and $L$ are finite complexes, then something like your co-Whitehead statement is true!


Theorem 1: If $f: K\to L$ is a map of finite complexes such that $\pi^k( \Sigma^n f)$ is an isomorphism for all $k\geq k\_0$ and all $n \geq n\_0$, then $\Sigma f$ is a homotopy equivalence. 


Corollary 2: In Theorem 1, if both $K$ and $L$ are simply-connected, then $f$ is a homotopy equivalence.


The proof uses a theorem of mine:


Theorem M: If $X$ is simply-connected and of finite type and $\mathrm{map}\_\*(X,S^k) \sim \*$ for all sufficiently large $k$, then $\mathrm{map}(X,Y)\sim \*$ for all finite-dimensional CW complexes $Y$.


Proof of Theorem 1: The hypotheses imply that the cofiber $C\_{\Sigma^{n\_0} f} \simeq \Sigma^{n\_0} C\_f$
satisfies $\mathrm{map}\_\*(\Sigma^{n\_0} C\_f, S^k) \sim \*$ for all $k \geq k\_0$. Theorem M implies that $\mathrm{map}\_\*(\Sigma^{n\_0} C\_f, \Sigma^{n\_0}C\_f) \sim \*$, which implies $\Sigma^{n\_0} C\_f \sim \*$ and hence that $\Sigma C\_f \sim \*$. This suffices to show that $\Sigma f$ is a homotopy equivalence.


