Understanding Vector Bundles Over Spheres

Question

I am trying to gain some familiarity with working with vector bundles and working on two problems that I found in the book Differential Topology by Hirsch. I have a feeling that the first problem is involved in the solution to the second problem, so I am asking these two questions together:

1) Given a $k$-plane bundle over $S^n$, by taking two charts that cover $S^n$ that contain the equator $S^{n-1}$ in there intersection, we obtain a map $S^{n-1} \to GL(k)$ via the transition map between the two charts. In this way we get a map from the set of $k$-bundles over $S^n$ to $\pi_{n-1}(GL(k))$. Hirsch asks to verify that this is a bijection.

It seems to me that this specifically involves proving that if two vector bundles have the same transition map up to homotopy along the equator, then they are the same. Additionally we need to show that any such map from $S^{n-1} \to GL(k)$ is realized as the transition map restricted to the equator as above.

2) Classify vector bundles over $S^1, S^2,$ and $S^3$. I imagine that this might follow from knowledge of homotopy groups of general linear groups via (1).

Thanks!

Answer 1

You can look at Hatcher's notes for 1). This is the so-called clutching construction.

$GL(k)$ has two components for $k>1$. Hence over every rank $k$ there are two vector bundles over $S^1$ (this is in bijection with [S^0,GL(k)] which has two components). The bundles are $\mathbb{R^k}$ and $\tau\oplus \mathbb{R}^{k-1}$ where $\tau$ is the tautological bundle over $\mathbb{RP}^1\cong S^1$.

It is a fact that $GL(1)=\mathbb{R}\setminus \{0\}$, $GL(2)\cong S^1$ and $\pi_1(GL(k))=\mathbb{Z}/{2\mathbb{Z}}$ for $k>2$. From this you obtain only a trivial line bundle, $\mathbb{Z}$ different plane bundles (can be obtained as tensor powers of the taulogical (complex) line bundle $\sigma$ on $\mathbb{CP}^1\cong S^2$. There are only two rank $k\geq 3$ bundles. They are the trivial one and $\sigma\oplus \mathbb{R}^{k-2}$.

Finally it turns out that $\pi_2(GL(k))=0$ for any Lie group, hence there are no non-trivial vector bundles over the three sphere.