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 $(n-1)$-connected manifolds of dimension $$2n$$ (since $$n$$-connected $$2n$$-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 $$V \to S^n$$ 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 $$S^{2n-1}$$, 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.