This question is related to this one but is not a duplicate. In " Characteristic Classes" by Milnor and Stasheff on pages 167-168, the authors give a brief argument about why the tangent bundle of the complex projective line $\tau_1$ is not isomorphic to its dual.
The argument says that if it were isomorphic, the isomorphism would map a tangent space to itself (and reverse orientation); one could then deduce the existence of a nowhere zero tangent line field on the sphere which is a contradiction. However, an isomorphism of vector bundles doesn't need to map a fiber to itself. It just should be a homeomorphism of the total space and map fibers isomorphically to fibers (page 14), or am I wrong?
Anyway, why isn't $\tau_1$ isomorphic to its dual?