I'm trying to understand Milnor's proof of the existence of exotic 7-spheres.

Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ).  Such a bundle can be described as follows:

Given $M$, an $S^{3}$ bundle over $S^{4}$, if we restrict $M$ to the northern (or southern) hemisphere of $S^{4}$, it must trivialize since each hemisphere is contractible.  Hence, we can build $M$ by specifying, for each point $p$ in $S^{3}$ = equator of $S^{4}$ = intersection of northern and southern hemispheres, an element of $SO(4)$ which glues $p\times S^{3}$ in the northern hemisphere to $p\times S^{3}$ in the southern hemisphere.

This defines a function $f:S^{3}\rightarrow SO(4)$, which is known as the clutching function for $M$.  By usual fiber bundle theory, the isomorphism type of $M$ only depends on the homotopy class of $f$.


$SO(4)$ is double covered by $S^3\times S^3$, and hence $\pi_3(SO(4)) = \mathbb{Z}\oplus\mathbb{Z}$.  Thus, $f$ is really determined (at least, up to homotopy) by an ordered pair of integers (i,j).


Now, as the bundles have structure group $SO(4)$, it makes sense to talk about the Pontryagin classes of $M$.  In Milnor's proof of the existence of exotic spheres, he needs to argue that $p_1(M) = \pm 2(i-j)$.  His first step in this argument is that "clearly $p_1(M)$ is a linear function of $i$ and $j$."


It IS clear to me that the Pontragin classes associated to $(ni, nj)$ for $n\in \mathbb{Z}$ will depend linearly on $n$.  For, if we let $N_{i,j}$ denote the principal $SO(4)$ bundle over $S^{4}$ corresponding to $(i,j)$, then $N_{ni,nj}$ is clearly obtained as the pullback of $N_{i,j}$ via a degree $n$ map from $S^{4}$ to itself.

However, it's not clear to me why $p_1(M)$ is additive in $(i,j)$.  Am I missing something simple?


And while we're talking about it, is more true?  That is, For any sphere bundle over a sphere, say, $S^{k}\rightarrow E\rightarrow S^{n}$, should any characteristic classes (Pontryagin, Stiefel-Whitney, Euler) be linear in terms of the clutching function?

For example, we can think of $p_1$ as a map from $\pi_{n-1}(SO(k+1))\rightarrow H^{4}(S^{n})$.  Is this map a homomorphism?  How about for the other characteristic classes?