There is an inclusion $O(n)\hookrightarrow U(n)$ which views an $n\times n$ orthogonal matrix as a unitary matrix. It is also a theorem, sometimes called Bott periodicity, that we have the following homotopy groups:
$\pi_{4i-1}(O(n)) = \mathbb{Z}$ and $\pi_{4i-1}(U(n))=\mathbb{Z}$
for $n$ large, say $n>2(4i +1)$. My question is: What is the induced map $\pi_{4i-1}(O(n))\rightarrow \pi_{4i-1}(U(n))$? It is a map from $\mathbb{Z}$ to $\mathbb{Z}$ so must be multiplication by some integer, but which integer is it?
The only thing I know that might help is that the map $O(n)\hookrightarrow U(n)$ induces a map on classifying spaces $BO(n)\rightarrow BU(n)$. One can think of this map as an inclusion of a real grassmannian into a complex grassmannian, but the map on homotopy groups seems just as mysterious.
Thanks for any help.