self-Whitney sum of the canonical vector bundle on Grassmannians

Question

Let $G_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle $$ \gamma_{k,N}: \mathbb{R}^k\longrightarrow E(\gamma_{k,N})\longrightarrow G_k(\mathbb{R}^N) $$ where \begin{eqnarray*} E(\gamma_{k,N})&=&\{(V,v)\mid V\in G_k(\mathbb{R}^N), v\in V\} \\&=&V_k(\mathbb{R}^N)\times_{GL(k)}\mathbb{R}^k. \end{eqnarray*} Here $V_k(\mathbb{R}^N)$ is the Stiefel manifold consisting of $k$-tuples or linearly independent vectors in $\mathbb{R}^N$. In particular, when $k=1$, we have the canonical Hopf line bundle $$ \gamma_{1,N}: \mathbb{R}\longrightarrow S^N\times_{ \mathbb{Z}/2} \mathbb{R}\longrightarrow \mathbb{R}P^N. $$

Question. Are there references/results for the order (the order of a vector bundle is the smallest positive integer $n$ such that the $n$-fold self-Whitney sum of the vector bundle is trivial) of $\gamma_{k,N}$?

In the paper Vector fields on spheres, J.F. Adams, 1962, Theorem 7.4, it is proved $$ \text{order}(\gamma_{1,N})=2^{\phi(N)} $$ where $\phi(N)$ is the number of positive integers no larger than $N$ and congruent to $0,1,2,4$ mod $8$. I want to find references/generalizations to $\gamma_{k,N}$.

Answer 1

Let me expand the comment above a bit. Consider the tautological bundle $\tau\to\mathbb C P^2$. It is complex and has total Chern class $c(\tau)=1+a$, where $a\in H^2(\mathbb C P^2)$ generates the cohomology ring of $\mathbb C P^2$. Viewed as a real bundle, it has $$p_1(\tau_{\mathbb R})=-c_2(\tau\oplus\bar\tau)=-c_1(\tau)c_1(\bar\tau)=a^2$$ because $\tau_{\mathbb R}\otimes_{\mathbb R}\mathbb C\cong \tau\oplus\bar\tau$. But $a^2$ generates $H^4(\mathbb C P^2)\cong\mathbb Z$, so $\tau_{\mathbb R}$ has order $\infty$.

So, as soon as $N_0$ is large enough such that $G_k(\mathbb R^{N_0})$ classifies $\tau_{\mathbb R}$, you get order $\infty$ for $\gamma_{2,N_0}$. By adding a trivial bundle to $\tau$, the same argument shows that the order of $\gamma_{2+n,N_0+n}$ is $\infty$ as well.