Let me first explain the background of my question.
As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration sequence
$$ SO(n) \to SO(n+1) \to S^n.$$
Indeed this fibration is a principal-$SO(n)$-bundle and can actually be identified with the frame bundle of the tangent bundle of $S^n$, i.e., the tangent bundle $TS^n$ is the associated bundle to this principal-$SO(n)$-bundle along the tautological representation of $SO(n)$ on $\mathbb{R}^n$.
Now let us consider the special case $n = 4$. Here we get a principal-$SO(3)$-bundle
$$ SO(3) \to SO(4) \to S^3.$$
But we can identify the $S^3$ with $Sp(1)$, the symplectic group acting on the quaternions. Via this identification $S^3$ sits canonically in $SO(4)$, i.e. we have an embedding of topological groups $S^3 \cong Sp(1) \to SO(4)$.
The standard theory of principal-bundles hence tells us that the above bundle splits (topologically) which means that there is a homeomorphism
$$ SO(4) \cong SO(3)\times S^3 $$
in particular there is a continuous map $SO(4) \to SO(3)$ (which is not a morphism of lie groups).
It is well known that the homotopy type $BSO(n)$ represents the functor of $n$-dimensional oriented vector bundles, and via the clutching function construction the group $SO(n)$ itself represents the functor of $n$-dimensional vector bundles over suspensions of spaces. Hence the above map $SO(4) \to SO(3)$ tells us that there is a natural transformation between the functor of rank $4$ vector bundles over suspensions to the functor of rank $3$ vector bundles over suspensions.
Does anybody know a geometric construction of this transformation which I only understand homotopy theoretically?
Any ideas or references are greatly appreciated.
Best Answer
The double cover $\pi:S^3\times S^3\rightarrow SO(4)$ actually provides two homotopoically distinct maps from $S^3 = Sp(1)$ to $SO(4)$, given by restricting $\pi$ to either factor. (One can easily see that these two maps induce different maps on $\pi_3$, so are not homotopic).
The following theorem is contained in
Ziller has a freely accessible version here - see the middle of page $8$.
Theorem If $E\rightarrow M$ is a rank $4$ vector bundle over $M$ (where $M$ is a compact simply connected manifold.), then $\Lambda^2(E) = \Lambda^2(E)_+\oplus\Lambda^2(E)_-$ decomposes into self-dual and antiself dual forms. Then $\Lambda^2(E)_{\pm}$ are the two rank $3$ vector bundles over $M$ corresponding to the two sections above.