Kervaire invariant theory (that is: the question of when a given framed manifold can be converted into a sphere via surgery) was dormant from 1969 until 2009, when Hill, Hopkins, and Ravenel announced a proof (published in 2016) of the Kervaire Invariant Theorem. They now have a wonderful book about this work, where I learned the quotations below.

After a flurry of activity in the 1950s and 60s, the problem was solved in all dimensions other than those of the form $2^{j+1}-2$. The Kervaire invariant of a manifold is always 0 or 1, and if it's 0 then there's no obstruction to framed surgery, so $M$ is framed cobordant to a homotopy sphere.

In dimensions of the form $2^{j+1}-2$, Browder (1969) proved that there exists a framed manifold of dimension $2^{j+1}-2$ with nontrivial Kervaire invariant if and only if the element $h_j^2$ in the Adams spectral sequence (at the prime 2) is a permanent cycle, i.e., survives to an element $\theta_j$ in $\pi_{2^{j+1}-2}^S$. It was known that such manifolds do exist for $j < 6$, i.e., in dimensions 2, 6, 14, 30, and 62. But there were infinitely many dimensions (126, 254, ...) where the answer wasn't known.

Just before the Hill, Hopkins, Ravenel announcement, Victor Snaith published a book with the following quotes:

As ideas for progress on a particular mathematics problem atrophy it can disappear. Accordingly I wrote this book to stem the tide of oblivion.

He also wrote:

In the light of the above conjecture and the failure over fifty years to construct framed manifolds of Arf–Kervaire invariant one this might turn out to be a book about things which do not exist. This [is] why the quotations which preface each chapter contain a preponderance of utterances from the pen
of Lewis Carroll.

In 2009, Hill, Hopkins, and Ravenel proved that, indeed, there are no manifolds of Kervaire invariant one in those dimensions (for $j \geq 7$, leaving the case $j=6$, corresponding to dimension 126, open). The field is no longer dormant. People are working hard to compute stable homotopy groups (especially, using new techniques from motivic homotopy) to answer this question in dimension 126. A 2017 Annals paper by Wang and Xu built on this story to prove there are no exotic smooth structures on spheres in dimensions 5, 6, 12, 56, and 61.

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