# A $\mathbb{R}^{n}$ -fiber bundle which do not admit a n-dimensional vector bundle structure

Is there a fiber bundle $(E,B, \mathbb{R}^{n})$, with typical fiber $\mathbb{R}^{n}$, such that there is no any $n$-dimensional vector bundle structure on the pair $(E,B)$? That is there is no a continuous map $p:E \to B$ such that the triple $(E,B,P)$ would be a $n$ dimensional vector bundle.

• You might want to look for the universal case $BTop(n)$ in the literature. – user95545 Oct 26 '16 at 14:05
• see "Open and Closed Disk Bundles" by William Browder, jstor.org/stable/1970428. – Igor Belegradek Nov 23 '16 at 4:02

Such a fibre bundle does not exists if you suppose that it is endowed with a differentiable structure. Stewart has shown that the group of diffeomorphisms of $R^n$ retract to $O(n)$. So every $Diff(R^n)$-bundle has an $O(n)$-reduction.

Stewart, T. E. (1960). On groups of diffeomorphisms. Proceedings of the American Mathematical Society, 11(4), 559-563.

• Thank you very much for your interesting answer and helpful reference. – Ali Taghavi Oct 29 '16 at 7:11

Yes, there a lots of them.

The topological tangent bundle $\tau$ of any topological non-smoothable manifold $M$ gives you an example since a vector bundle reduction of $\tau$ is by smoothing theory equivalent to a smoothing of $M$.

• Thank you very much for your interesting answer. Unfortunately I can not accept two answers simultaneously. – Ali Taghavi Oct 29 '16 at 7:13