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Let $E\to M$ be a vector bundle. Is the structure of vector bundle determined by the map $E\to M$ (as morphism between manifolds)? i.e. is it possible that there are two non-isomorphic vector bundles $E_1, E_2$ such that their underlying manifolds are the same?

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  • $\begingroup$ If you are willing to consider complex vector bundles, you can get examples more easily (although I'm sure you can also do it with only real vector bundles). For example, the tautological line bundle and its (complex) dual are isomorphic as real vector bundles and so in particular their total spaces are diffeomorphic but they are not isomorphic as complex vector bundles. $\endgroup$ – levap Dec 2 '17 at 22:45
  • $\begingroup$ @levap Yeah. But in that case I would require the total spaces to be holomorphic equivalent. My point is, when talking about vector bundles, can we only mention the map (in the corresponding category) $E\to M$ (and forget about the local trivialization)? $\endgroup$ – Akatsuki Dec 2 '17 at 23:15
  • $\begingroup$ I came up with this question when I try to understand the definition of some specific vector bundles. For example in some books the definition of tautological bundles only mentions the bundle projection "in the tautological way" and does not mention the local trivializations. I know in this case the fibre itself has the linear structure so we don't have to define it, but I still want to know in general is it true that we don't need to mention the linear structure of every fibre $\endgroup$ – Akatsuki Dec 2 '17 at 23:35
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This answer is close, but not quite what you are asking: Here is a countably infinite family of examples $E_i$ of distinct vector bundles over $S^4$ which are homeomorphic as abstract manifolds. I don't know if any pair of them are diffeomorphic or not.

By clutching functions, there are a $\pi_3(SO(4))\cong\mathbb{Z}\oplus\mathbb{Z}$s worth rank $4$ bundles over $S^4$. From Milnor's exotic $7$-spheres paper, we know that a countable subfamily of these has Euler class $\pm 1$. Let $E_i$ enumerate these.

Using the notation $SE_i$ for the sphere bundle in $E_i$, we note that $SE_i$ is an $S^3$ bundle over $S^4$ with Euler class $1$. The Gysin sequence then shows that $SE_i$ is a homotopy $S^7$, which must then be homeomorphic to $S^7$ by the Poncare conjecture. In particular, $SE_i$ is homeomorphic to $SE_j$.

Now, one can use the Alexander trick to extend such a homeomorphism to a homeomorphic from $E_i$ to $E_j$, so these manifolds are pairwise homeomorphic, even though they are different vector bundles.

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