Michael J. Hopkins
Higher real K-theories and topological automorphic forms.
Tue, 10/06/2009 - 17:01 — SomNambulAuthors: Michael J. Hopkins, Mark Behrens
Subjects: Algebraic Topology
AbstractGiven a maximal finite subgroup G of the nth Morava stabilizer group at a
prime p, we address the question: is the associated higher real K-theory EO_n a
summand of the K(n)-localization of a TAF-spectrum associated to a unitary
similitude group of type U(1,n-1)? We answer this question in the affirmative
for p in {2, 3, 5, 7} and n = (p-1)p^{r-1} for a maximal finite subgroup
containing an element of order p^r. We answer the question in the negative for
all other odd primary cases.On the non-existence of elements of Kervaire invariant one.
Thu, 08/27/2009 - 13:23 — SomNambulAuthors: Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel
Subjects: Algebraic Topology
AbstractWe show that Kervaire invariant one elements in the homotopy groups of
spheres exist only in dimensions at most 126. By Browder's Theorem, this means
that smooth framed manifolds of Kervaire invariant one exist only in dimensions
2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this
resolves a longstanding problem in algebraic topology.
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