The smooth structure determined by \( (\mathbb{R}^n, \mathrm{id}_{\mathbb{R}^n}) \) on \( \mathbb{R}^n \) is called the standard smooth structure.
Remark
- For\( n \neq 4 \), every smooth structure on \(\mathbb{R}^n\) is diffeomorphic to standard one.
- For \(n=4\)
- For \(\mathbb{R}^4\), there exists uncountably many smooth structures which are not diffeomorphic to each other (Donaldson-Freedman, 1984). The Euclidean space \(\mathbb{R}^4\) equipped with a nonstandard structure is called fake \(\mathbb{R}^4\).