TITLE: Whitehead theorem for cohomotopy
QUESTION [10 upvotes]: 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:

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.

Does the following conjecture hold?

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.

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 $H^k(M)$.

REPLY [10 votes]: Let me just add that stably it is true that a map f: $X\xrightarrow{} Y$ between finite complexes that induces isomorphisms on cohomotopy (of all degrees, now, including negative) is a weak equivalence.  This is because the collection of all Z such that 
$f^{*} : [Y,Z] \xrightarrow{} [X,Z]$
is an isomorphism is a thick subcategory (also closed under products)
so if it contains the spheres it contains all finite complexes.  It is not true for general Y and Z because there are many many spectra with no cohomotopy at all; the Morava K-theories for example.  
This also points to what goes wrong unstably, because mapping out of things behaves well with respect to fibrations, not cofibrations.  So you'd expect the co-Whitehead theorem to be true not for finite complexes, but for spaces that can be built from spheres by taking iterated total spaces of fibrations; Lie groups for example, maybe.