The space of homotopy classes of maps of products of spheres Proposition 17.6.1 of "Differential form in Algebraic Topology" by Bott and Tu proves the following beautiful result:
$[S^{q}, X]\simeq \frac{\pi_{q}(X,x)}{\pi_{1}(X,x)}$
where $S^{q}$ is the $q$-sphere, $X$ is any path connected space and $[S^{q}, X]$ is the set of homotopy classes of continuous maps from $S^{q}$ to $X$. I was wondering if there was an analogous result for the product of spheres, namely:
$[S^{q}\times S^{p}, X] = ?$
I am mostly interested in the cases where $S^{q}$ and $S^{p}$ are both parallelizable and $X$ is the classifying space for $U(k)$, namely $X = BU(k)$. In other words, I am interested in the set of equivalence classes of complex vector bundles of rank $k$ on $S^{q}\times S^{p}$.
Thanks.
 A: This is not a complete answer, but is intended to expand on some of the comments above.
As mentioned by j.c. and Dylan Wilson in the comments, the set of unbased homotopy classes $[S^p\times S^q,X]$ is a quotient of the set of based homotopy classes $\langle S^p\times S^q,X\rangle$ under an action of $\pi_1(X)$. Since you say you are mostly interested in the case $X=BU(k)$, which is simply-connected, we may as well work with pointed homotopy classes.
The way I would try to analyse this (and perhaps this is the approach Ryan Budney had in mind in his comment) is using the cofibration sequence
$$
S^{p+q-1}\to S^p\vee S^q\to S^p\times S^q \to S^{p+q}\to S^{p+1}\vee S^{q+1}\to \cdots
$$
in which the first map is the attaching map of the top cell of $S^p\times S^q$. The fourth map is the suspension of this attaching map, therefore is null-homotopic. There results an exact sequence of pointed sets
$$
1 \to \langle S^{p+q},X\rangle \to \langle S^p\times S^q,X\rangle \to \langle S^p\vee S^q,X\rangle \to \langle S^{p+q-1},X\rangle
$$
which written in terms of homotopy groups becomes
$$
0\to \pi_{p+q}(X) \to \langle S^p\times S^q,X\rangle \to \pi_p(X)\oplus\pi_q(X)\to \pi_{p+q-1}(X).
$$
The last map is the Whitehead product. Hence if you know the homotopy groups of $X$ (which for $X=BU(k)$ are known in a stable range by Bott periodicity) and the Whitehead products, you have a good chance of describing the set $\langle S^p\times S^q, X\rangle$ (which by the way is not a group, since $S^p\times S^q$ is not a co-H-space).
Edit: In the comments, user DLIN asks about the case $X=Gl_N(\mathbb{C})$. We can say more in this case, since $X$ is a path-connected H-space. Therefore all Whitehead products in $\pi_*(X)$ are trivial, and the exact sequence of sets above becomes a short exact sequence of groups
$$
0\to \pi_{p+q}(X) \to \langle S^p\times S^q,X\rangle \to \pi_p(X)\oplus\pi_q(X)\to 0.
$$
Note that the middle term is now a group (using an H-space multiplication $\mu:X\times X\to X$). Furthermore, this sequence splits; the last map, which is given by restriction to the two sphere factors, is split by sending $(f,g)$ to $\mu\circ (f\times g)$.
Now suppose we know that $\langle S^p\times S^q, X\rangle$ is abelian. Then we could conclude that
$$
\langle S^p\times S^q, X\rangle \cong \pi_{p+q}(X)\oplus\pi_p(X)\oplus\pi_q(X).
$$
This would be the case, for example, if there are two commuting H-space structures on $X$ (see the Eckmann-Hilton argument).
