Milnor's sphere
In mathematics, specifically differential and algebraic topology, Milnor's sphere is the first discovered exotic sphere. During the mid 1950's John Milnorpg 14 was trying to understand the structure of -connected manifolds of dimension 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 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, 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.