TITLE: Manifolds with polynomial transition maps
QUESTION [21 upvotes]: Title edited I  thank  მამუკა ჯიბლაძე  and  Corbennick for  their  suggestion on the  title  of this  question. I  changed the  title based on the suggestion of  Corbennick.
What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:
For every two charts $(\phi,U)$ and $(\psi,V)$ in $\mathcal{A}$, $\psi \circ \phi^{-1}$ is a polynomial map.(Its  components are  polynomial  functions).
Motivation: The  motivation for this  question comes  from the  concept "Affine  manifolds". I  am  indebted to Mike  Cocos,  for  learning this concept and its  related problem, that is  Chern  conjecture.

REPLY [21 votes]: Brief sketch of slight simplification of Bryant's answer:
Without loss of generality $\mathcal A$ is maximal with respect to the condition that all transition functions are polynomial. Now make a space $\tilde M$ by gluing together $U$ and $V$ whenever $(\phi, U)$ and $(\psi,V)$ are charts such that $\phi$ and $\psi$ agree in $U\cap V$. This is a smooth manifold (or rather each of its connected components is), because polynomials are so rigid. It is equipped with both a covering projection to $M$ and an immersion in $\mathbb R^n$. So if $M$ is simply connected then $M$ immerses in $\mathbb R^n$. If $M$ is also compact and $n>0$ then this is impossible.