Many topological spaces have canonical or “obvious” smooth structures. For instance a Cartesian space has the evident smooth structure induced from the fact that it can be covered by a single chart – itself.
From this example, various topological spaces inherit a canonical smooth structure by embedding. For instance the -sphere may naturally be thought of as the collection of points
given by and this induces a smooth structure of .
But there may be other, non-equivalent smooth structures than these canonical ones. These are called exotic smooth structures. See there for more details.
John Stallings, The piecewise linear structure of Euclidean space , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. (pdf)
Revised on May 11, 2013 15:39:22