### ‘Kervaire Invariant One Problem’ Solved

#### Posted by John Baez

Big news! It seems Mike Hopkins, Doug Ravenel and Mike Hill have cracked the Kervaire Invariant One problem.

Hopkins announced this in a maximally dramatic fashion, as explained below…

Here’s what Nick Kuhn wrote on the ALGTOP mailing list:

Yesterday, at the conference on Geometry and Physics being held in Edinburgh in honor of Sir Michael Atiyah, Harvard Professor Mike Hopkins announced a solution to the 45 year old Kervaire Invariant One problem, one of the major outstanding problems in algebraic and geometric topology. This is joint work with Rochester professor Doug Ravenel and U VA postdoctoral Whyburn Instructor Mike Hill.

The solution completes the work on ‘exotic spheres’ begun by John Milnor in the 1950’s which led to his Fields Medal. This is a central part of the classification of manifolds (= curves, surfaces, and their higher dimensional analogues). A 1962 Annals of Math paper by Milnor and Michael Kervaire classified exotic differential structures on spheres, subject to one possible ambiguity of order 2 in even dimensions. A 1969 Annals Math paper by Princeton professor William Browder resolved this question, except when the dimension was 2 less than a power of 2. In these dimensions, he translated the problem into one in algebraic topology, specifically one about the existence of certain elements in the stable homotopy groups of spheres. Over the next decade, the elements in dimensions 30, 62, and 126 were shown to exist; equivalently there exist some manifolds in those dimensions with some oddball properties. Significant work on closely related problems was done by Northwestern professor Mark Mahowald.

So yesterday’s announcement was that in all higher dimensions (254, 510, 1022, etc.), the putative elements do NOT exist. This result is ‘detected’ in a generalized homology theory that is periodic of period 256 built from the complex oriented theory associated to deformations of the universal height 4 formal group law at the prime 2. (By contrast, real K-theory is has period 8 and comes from height 1 deformations, and theories based on elliptic cohomology come from height 2.) The strategy of proof has similarity to work of Ravenel’s from the late 1970’s, but the success of the strategy now illustrates the power of newly emerging control of subtle number theoretic and group theoretic structure in algebraic topology.

(2) Technical stuff, which may or may not be accurate …

Step 1. Using results/methods from Miller, Ravenel and Wilson, one can show if $\Theta_j$ is nonzero, then it is nonzero in $\pi_*(E_4^{hZ/8})$, for some well chosen action of $Z/8$ on the 4th 2-adic Morava E theory.

Step 2. Using a spectral sequence associated to a cleverly chosen filtered equivariant model for $E_4$ (or similar ??) - and this is the very new bit, I think - one shows that

(a) $\pi_{-2}(E_4^{hZ/8}) = 0$ and

(b) $\pi_*(E_4^{hZ/8})$ is 256 periodic.

Thus the $\Theta_j$’s cannot exist beginning in dimension 254.

Café regulars should be tantalized to hear that Hopkins’ argument involves invariants that live two steps higher in the ‘chromatic filtration’ than elliptic cohomology. If you have no idea what that means, try week197 for a gentle introduction. But if even that’s too tough, here’s the idea in even more simplified form.

$K$-theory is an invariant of topological spaces that’s built using vector bundles. In quantum theory we use vector bundles to study how *point particles* move around under the influence of forces called ‘gauge fields’, like electromagnetism .

Elliptic cohomology is an invariant of topological spaces that’s one step higher in a sequence called the ‘chromatic filtration’ — it’s ‘height two’. It began life as a piece of very abstract homotopy theory — which Mike Hopkins was instrumental in developing. But thanks to some brilliant work by people including Ed Witten, Graeme Segal, and more recently Stephan Stolz and Peter Teichner, it’s becoming clear that this math is related to the physics of *strings*. The worldsheet of a string has dimension one higher than the worldline of a particle.

So, while I bet it’s not yet visible in Hopkins’ work, I bet that the invariant Hopkins uses — the 4th 2-adic Morava $E$ theory — will turn out to be at least slightly related to theories of *3-branes*.

(If this guess is completely stupid, please explain why.)

## Re: ‘Kervaire Invariant One Problem’ Solved

My personal spy on that conference, Danny Stevenson, mentioned, if I understood him correctly, that Hopkins also said something about the existence of “generalized Clifford algebras” in some of these dimensions. Er, maybe I don’t know in which dimensions, is it these 254, 510, 1022… and then maybe 34560?

Anyway, I am guessing this must be related to (an even higher version of) the higher Clifford algebras discussed here?

One kind of higher Clifford algebra per chromatic filtration/brane dimension?