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 [MillerRavWilson], 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.