Milnor's sphere explained

In mathematics, specifically differential and algebraic topology, during the mid 1950's John Milnor^{[1] pg 14} was trying to understand the structure of

(n -1 ) -connected manifolds of dimension

2 n (since n -connected 2 n -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.

See also

• Exotic sphere
• Oriented cobordism

Notes and References

1. Web site: Ranicki. Andrew. Roe. John. Surgery for Amateurs. live. https://web.archive.org/web/20210104182942/https://sites.psu.edu/surgeryforamateurs/files/2017/12/surgerybook2017-2gfid7m.pdf. 4 Jan 2021.