Non-(stable)-triviality of the tautological bundles

Question

This is a question I asked at Math.SE but got no answers: https://math.stackexchange.com/q/396217/7110/

The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold $G_k(\mathbb{K}^N)$ of all $k$-planes in $\mathbb{K}^N$ (for $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$) is defined as $\gamma_k(\mathbb{K}^N) := \{(x, v) \in G_k(\mathbb{K}^N) \times \mathbb{K}^N: v \in x\}$. I wondered whether any of those bundles is (stably) trivial:

For which $k$, $N$ is $\gamma_k(\mathbb{K}^N)$ (stably) trivial?

Sure it is the case for $k = 0$ or $k = N$ since then $G_k(\mathbb{K}^N)$ is just a point. Furthermore, I know that $\gamma_1(\mathbb{R}^2)$ is not stably trivial since it is the Möbius bundle which has non-trivial first Stiefel-Whitney class $\omega_1(\gamma_1(\mathbb{R}^2)) \not= 0$. Similar, since $c_1(\gamma_1(\mathbb{C}^N)) \not= 0$ (more or less per definition), we know that $\gamma_1(\mathbb{C}^N)$ is not stably trivial.

I think that $\gamma_k(\mathbb{K}^N)$ is never stably trivial unless $k = 0, N$. But I don't know how to prove this. Maybe by a more elaborate argument using the Stiefel-Whitney or Chern classes? But then, what about $\gamma_k(\mathbb{H}^N)$, which do not have associated characteristic classes (as far as I know)?

Added: What about the tautological bundles $\gamma_k(\mathbb{O}^N)$ for $k = 1$ and $N = 2, 3$?

Answer 1

For simplicity, let's take $\Bbb K = \Bbb R$.

By the bundle classification theorem, your question amounts to understanding whether the inclusion map

$$ G_k(\Bbb R^N) \to \underset j{\text{colim }} \, G_{k+j}(\Bbb R^{N+j}) = BO $$ is null homotopic.

First consider the inclusion $$ i: G_k(\Bbb R^N) \to G_k(\Bbb R^\infty) = BO_k \, . $$

According to Milnor and Stasheff (page 81), the restriction homomorphism $$ i^* : H^p(BO_k) = H^p(G_k(\Bbb R^\infty)) \to H^p(G_k(\Bbb R^N)) $$ (with any coefficients) is an isomorphism in degrees $p < N-k$. Since $H^p(BO_k;\Bbb Z_2)$ is a polynomial algrbra on the Stiefel-Whitney casses $w_1,\dots,w_k$, it follows that $i^*$ is not trivial in degrees $p \le N-k$.

On the other hand, also by Milnor and Stasheff, the restriction homomorphism $$ H^p(BO;\Bbb Z_2) \to H^p(BO_k;\Bbb Z_2) $$ is an isomorphism in degrees $p \le k$. It follows that the homomorphism $$ H^p(BO;\Bbb Z_2) \to H^p(G_k(\Bbb R^N);\Bbb Z_2) $$ is not trivial for all $p$ such that $0 < p \le \min(N-k,k)$. In particular, this is true for some $p >0$ whenever $0 < k < N$.

So the answer to your question is no when $0 < k< N$.

A similar argument works for the other $\Bbb K$.