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Given a smooth map between compact manifolds without boundary what criteria guarantee that the induced bundle is isomorphic to the tangent bundle? And less generally, suppose the map has non-zero Brouwer degree. What then?

Joe S
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    What do you mean by induced bundle? – Jesus RS Apr 16 '15 at 22:09
  • The pull back bundle. I have thought more on this and realize this is no easy question. Examples where this works are the Gauss map for hyper surfaces, and coverings. The tangent bundle is the induced from its classifying map into a Grassmann manifold. For maps of a manifold into itself, only maps of degree +/-1 can work unless the Euler class is zero. – Joe S Apr 21 '15 at 14:23
  • That's the thing. For an example (very easy) of the use of homology and pullbacks, see here http://math.stackexchange.com/questions/1195400/the-bundle-vector-f-ast-xi-for-moebius-over-s1/1217041#1217041. What follows from this argument is that maps $f:S^1\to S^1$ always have the property. But I guess from your comment that you know this and how to work more general situations. – Jesus RS Apr 21 '15 at 18:02

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