Changes

===Homeomorphism v. diffeomorphism===

===Homeomorphism v. diffeomorphism===

For those readers who have some experience with topology, there may be a question of whether a diffeomorphism always exists.  While the existence of a homeomorphism between manifolds demonstrates the existence of a diffeomorphism between them if they are of dimension <math>\leq 3 \ </math>, and it turns out that a diffeomorphism exists for <math>n=5 \ </math> and <math>n=6 \ </math>, but for most <math>n\geq 7 \ </math> it is possible to construct objects which are homeomorphic and not diffeomorphic - such objects are called "exotic <math>\mathbb{S}^n</math>." The only open case is the four dimensional one.

For those readers who have some experience with topology, there may be a question of whether a diffeomorphism always exists.  While the existence of a homeomorphism between manifolds demonstrates the existence of a diffeomorphism between them if they are of dimension <math>\leq 3 \ </math>, and it turns out that a diffeomorphism exists for <math>n=5 \ </math> and <math>n=6 \ </math>, but for most <math>n\geq 7 \ </math> it is possible to construct objects which are homeomorphic and not diffeomorphic - such objects are called "exotic <math>\mathbb{S}^n</math>." The only open case is the four-dimensional one.

==Proof of the conjecture==

==Proof of the conjecture==