Milnor's sphere

In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnor[1]^{pg 14} was trying to understand the structure of ${\displaystyle (n-1)}$-connected manifolds of dimension ${\displaystyle 2n}$ (since ${\displaystyle n}$-connected ${\displaystyle 2n}$-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 ${\displaystyle V\to S^{n}}$ 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 ${\displaystyle S^{2n-1}}$, 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.