It is fairly known that given a cover $\{U_i\}_{i \in \mathbb{N}}$ of some smooth manifold $M$ together with smooth transition functions $g_{\alpha \beta} \colon U_{\alpha} \cap U_{\beta} \rightarrow GL_n(\mathbb{R})$ satisfying the cocycle condition $g_{\alpha \beta} g_{\beta \gamma} = g_{\alpha \gamma}$ this defines a smooth vector bundle of rank $n$ on $M$ (upto isomorphism).

Say given another set of such transition functions $h_{\alpha \beta}$ together with homotopies (which are smooth transition functions for every $t$) from the $g_{\alpha \beta}$ to the $h_{\alpha \beta}$ is it true that the smooth vector bundles defined by the $g$ and $h$ are diffeomorphic?

I am aware of such a fact for topological vector bundles, for example Thm. 4.3 and corollaries in Hunsemöller, Fibre Bundles. But I am not convinced that the construction he gives there would be smooth. At least he uses some maximum function (however, I find the proof quite difficult to read and would be happy to see another version of it by another author, even if it should not work for differential bundles).

  • $\begingroup$ I must be missing something. Why not cover $M$ with small contractible open sets $U_i$ so that $U_i\cap U_j$ are all contractible? $\endgroup$ – Ted Shifrin Nov 12 '14 at 1:13
  • $\begingroup$ Of course you can create the Mōbius strip by gluing on small intervals. $\endgroup$ – Ted Shifrin Nov 12 '14 at 12:13
  • $\begingroup$ That said, since $GL(k,\Bbb R)$ has two components, there's no hope to have homotopic transition functions unless both bundles are orientable or non-orientable. $\endgroup$ – Ted Shifrin Nov 12 '14 at 12:43

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