# Does a covering by trivializations imply existence of a linearly compatible one, for a smooth $\mathbb{R}^k$ fiber bundle?

Let $$\pi : E\rightarrow B$$ be a surjective submersion between smooth submanifolds. Let $$k=\mathrm{dim}\,\mathrm{ker}\,\pi_*$$.

We define a local trivialization of $$\pi$$ with fiber $$\mathbb{R}^k$$ to be a pair $$(U,\phi)$$ where $$U\subset B$$ is open and $$\phi : \pi^{-1}(U) \xrightarrow{\approx} U\times \mathbb{R}^k$$ is a diffeomorphism such that $$\phi(y) \in \{\pi(y)\} \times \mathbb{R}^k \;\forall y\in \pi^{-1}(U)$$.

We define an atlas of trivializations to be a collection $$\{(U_\alpha,\phi_\alpha)\}_\alpha$$ where each $$(U_\alpha,\phi_\alpha)$$ is a local trivialization and the $$U_\alpha$$'s cover $$B$$.

We say an atlas of trivializations is linearly compatible if for every indices $$\alpha,\beta$$, for every $$p \in U_\alpha \cap U_\beta$$, the map $$\mathbb{R}^k \rightarrow \{p\} \times \mathbb{R}^k : v \mapsto \phi_\alpha(\phi_\beta^{-1}(p,v))$$ followed by $$\{p\} \times \mathbb{R}^k \rightarrow \mathbb{R}^k : (p,w) \mapsto w$$ is $$\mathbb{R}$$-linear.

My question is, if $$\pi$$ admits an atlas of trivializations with fiber $$\mathbb{R}^k$$, then does it also admit a linearly compatible atlas?

Edit: as has been pointed out in the comments, this fails if we relax $$E$$ and $$B$$ to topological manifolds and "smooth"/"diffeo" to "continuous"/"homeo". I've now changed the question to focus on the smooth case.

• Are $E$ and $B$ differentiable manifolds? May 12 at 17:44
• I guess that is the case I'm most interested in; I'll edit the question to reflect that, though I'd still also like to hear about the general case. May 12 at 17:49
• So in the case of manifolds this translates to the question whether every topological manifold has PL structure, right? PL manifold is even weeker (then the one in the question) property, since transition maps are assumed to be piecewise linear (hence the name). The answer is negative in dimension $\geq 5$. May 12 at 18:01
• I see, thank you! However, does this also fail even if $E$ and $B$ are smooth manifolds, and $\pi$ is smooth and the trivializations are all diffeomorphisms? May 12 at 21:00
• @IndraneelTambe yes. Top, smooth and PL are distinct in dim $\geq 5$. May 13 at 5:20