Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
Posted this question on MO: mathoverflow.net/questions/131340 –  AlexE May 21 '13 at 13:37

1 Answer 1

up vote 2 down vote accepted

This question has been asked and answered on MathOverflow. I have replicated the accepted answer by John Klein below.

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 $\Bbb K$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.