John Milnor and His Spherical Exotica

John Milnor’s discovery of exotic smooth spheres unleashed the exploration of new mathematical riches. He was just awarded the 2011 Abel prize. Credit: MFO via Creative Commons License

On March 23, the Norwegian Academy of Science and Letters announced that it was awarding the 2011 Abel Prize — Norway’s mathematical answer to Sweden’s Nobels — to John Milnor. The Stony Brook mathematician, 80, had already won a Fields Medal and nearly every other major math prize there is.

As most math grad students have done over the past half-century — at least those students specializing in topology and geometry — when I was in grad school I learned much of what I know from Milnor’s canonized textbooks. (The most standard ones are probably “Characteristic Classes,” which he co-authored with James Stasheff, and “Morse Theory,” but my favorite one was “Lectures on the h-Cobordism Theorem,” a typewritten gem that has long been out of print but can be found online if you google it. Some day I’ll blog about why I liked it so much.)

And perhaps as many other topology students have done over the years, I have often tried to imagine what it must have felt to be a topologist in 1956, when Milnor announced his discovery of a 7-dimensional sphere that was homeomorphic, but not diffeomorphic, to the standard 7-dimensional sphere.

What does that mean? I have tried my best to explain the basic idea in my Scientific American article on the solution of the Kervaire invariant problem and then again in the article I co-wrote with my colleague John Matson on the announcement of Milnor’s Abel prize for


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