Milnor's sphere

In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnor^{pg 14} was trying to understand the structure of [math]\displaystyle{ (n-1) }[/math]-connected manifolds of dimension [math]\displaystyle{ 2n }[/math] (since [math]\displaystyle{ n }[/math]-connected [math]\displaystyle{ 2n }[/math]-manifolds are homeomorphic to spheres, this is the first non-trivial case after) and found an example of a space which is homotopy equivalent to a sphere, but was not explicitly diffeomorphic. He did this through looking at real vector bundles [math]\displaystyle{ V \to S^n }[/math] over a sphere and studied the properties of the associated disk bundle. It turns out, the boundary of this bundle is homotopically equivalent to a sphere [math]\displaystyle{ S^{2n-1} }[/math], but in certain cases it is not diffeomorphic. This lack of diffeomorphism comes from studying a hypothetical cobordism between this boundary and a sphere, and showing this hypothetical cobordism invalidates certain properties of the Hirzebruch signature theorem.

See also

• Exotic sphere
• Oriented cobordism