Milnor's sphere

{{refimprove|date=January 2021}}

In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnor{{Cite web|last=Ranicki|first=Andrew|last2=Roe|first2=John|date=|title=Surgery for Amateurs|url=https://sites.psu.edu/surgeryforamateurs/files/2017/12/surgerybook2017-2gfid7m.pdf|url-status=live|archive-url=https://web.archive.org/web/20210104182942/https://sites.psu.edu/surgeryforamateurs/files/2017/12/surgerybook2017-2gfid7m.pdf|archive-date=4 Jan 2021|access-date=|website=}}^{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 $V\; \backslash to\; S^n$ 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.