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This is a homework assignment. I'm not sure I even understand the question fully, as the parametrization seems slightly wrong.

Over an $n$-ball, let $r \in [0,1)$ denote a radial coordinate, and let $\theta$ denote the $n-1$ angular variables. One can build an $n$ sphere by glueing two balls together $(r_1 = 1- r_2, \theta_1 = \theta_2$, over $r_i \in (1/2, 1)$).

Assume that one can trivialise any vector bundle over both n-balls; the transition function, in the coordinates on the ball $B_1$, is a function $T(r,\theta), r\in (1/2,1)$.

a. Show that one can change the trivialisations so that the function is independent of r: the result can then be written as $T(\theta)$, a function on the $n-1$ sphere. b. Let $T, T'$ be two transition functions which are homotopic to each other: $T(\theta) = F(0,\theta), T'(\theta) = F(1,\theta)$ for $F\colon [0,1] \times S^{n-1} \rightarrow Gl(k,\mathbb{R})$. Show that the bundles they define are isomorphic.

c. Let $n> 1$. Show that one can arrange for $T, T'$ to lie in $Gl(k,\mathbb{R})^+$, the (connected) subgroup of matrices with positive determinant, by changing one of the trivialisations. Then show that if the two bundles defined by $T', T'$ are isomorphic, the transition functions are homotopic in $Gl(k, \mathbb{R})^+$, so that the bundles are classified by homotopy classes of maps $S^{n-1} \rightarrow Gl(k,\mathbb{R})^+$. What does this tell us about the classification of bundles over $S^2$?

So, I'm not sure I totally understand the glueing map, and for the purposes of the question I'm simply assuming that for $n=2$, my open balls are attached at a band along the equator of the 2-sphere (can't attach along boundary since they're open). I think the only important thing for c. is that for n > 1, we have that the intersection of $B_1$ and $B_2$ is connected, and so we have a homotopy.

Besides that, I've thought about construction some expression that is like the given transition function $T(r,\theta)$, but whose derivative with respect to $r$ vanishes, so that it would be constant and thus independent of $r.$ Otherwise, I can see myself using $b.$ to argue that for $r \in [0,1)$, we simply have a path in a connected component of $Gl_k$, so by homotopy of the transition functions we have an isomorphism of bundles... but the transition function should be different at every point of $r$, so I don't think that really works.

For b., I guess we have to expressly construct a diffeomorphism. The vector bundle structure defined by given transition maps is defined in Lee's Smooth Manifolds, p. 108 -- I don't know if it's a common construction, so I can include it if people are interested. It seems you can get from $E$, the smooth vector bundle defined by $T$ to $E'$ by composing a series of diffeomorphisms. e.g., if $\phi_1 \colon E \rightarrow B_1 \times R^k$ is a trivialisation, then $\phi_1 \circ \phi_2^{-1} (\theta, v) = (\theta,T(\theta)v) = (\theta,F(0,\theta)v) = (\theta,F(1,\theta),v) = (\theta,T'(\theta)v) = \phi_2'^{-1} ... $ if that makes no sense, no worries.

c. It makes sense to me that one can deformation retract $Gl_k^+$ to O(n) via gram-schmidt, but I don't really see how to construct the isomorphism or make this rigorous.

If anyone has any ideas, I'd be very appreciate. In particular, even rephrasing or explaining what the question is asking would be useful. As I'm sure is apparent, I don't have much of an idea of what's going on here :)

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