Classification of Vector Bundles over Spheres

[zhentil]

This is a question from Hirsch's Differential Topology book: show that there is a bijective correspondence between
$$K^k(S^n) \leftrightarrow \pi_{n-1}(GL(k))$$,
where $$K^k(S^n)$$ denotes the isomorphism classes of rank k vector bundles over the sphere. The basic idea is that any vector bundle over the sphere has a trivializing cover consisting of two open sets diffeomorphic to $$\textbf{R}^n$$. The transition function of such a cover restricts to a map from the equator $$S^{n-1} \rightarrow GL(k)$$. Moreover, vector bundles with homotopic classifying maps are isomorphic. However, the reverse inclusion is eluding me. Why must two isomorphic bundles over the sphere have homotopic classifying maps?

I've seen a proof of this involving writing the Grassmanian as a fiber bundle of orthogonal groups, and using the exact homotopy sequence to conclude that $$\pi_n(G_{3k,k}) \cong \pi_{n-1}(GL(k))$$, but since Hirsch never mentioned that, it strikes me that there must be an elementary way of seeing this. Any ideas?

 

[jvicens]

Hello,

Thank you for bringing up this interesting question from Hirsch's Differential Topology book. The proof you mentioned involving the Grassmanian as a fiber bundle of orthogonal groups is indeed one way to show the bijective correspondence between K^k(S^n) and \pi_{n-1}(GL(k)). However, there is indeed a more elementary approach that does not involve the use of the Grassmanian.

Firstly, let's recall the definition of the classifying space for vector bundles over a topological space X. This is a space BG such that there exists a bijective correspondence between isomorphism classes of rank k vector bundles over X and homotopy classes of maps from X to BG. In our case, we are interested in the classifying space for vector bundles over the sphere S^n, which is denoted by BS^n.

Now, let's consider the trivializing cover of a rank k vector bundle over S^n as mentioned in the forum post. This cover consists of two open sets diffeomorphic to \textbf{R}^n, and the transition function between these two open sets can be thought of as a map from the equator S^{n-1} to GL(k). This map is a homotopy classifying map for our vector bundle, since it captures the topological information of the bundle over the equator.

Now, suppose we have two isomorphic bundles over S^n. This means that there exists a diffeomorphism between the two bundles, which induces a homotopy equivalence between the corresponding classifying maps. This is because the diffeomorphism preserves the topological information of the bundle over each point on the equator. Therefore, the two bundles have homotopic classifying maps, and we have shown the reverse inclusion.

I hope this helps in understanding the bijective correspondence between K^k(S^n) and \pi_{n-1}(GL(k)). Please let me know if you have any further questions or if you would like me to elaborate on any specific part of the proof. Thank you.