37 research outputs found
The triviality of the 61-stem in the stable homotopy groups of spheres
We prove that the 2-primary is zero. As a consequence, the
Kervaire invariant element is contained in the strictly defined
4-fold Toda bracket .
Our result has a geometric corollary: the 61-sphere has a unique smooth
structure and it is the last odd dimensional case - the only ones are and .
Our proof is a computation of homotopy groups of spheres. A major part of
this paper is to prove an Adams differential . We prove this
differential by introducing a new technique based on the algebraic and
geometric Kahn-Priddy theorems. The success of this technique suggests a
theoretical way to prove Adams differentials in the sphere spectrum inductively
by use of differentials in truncated projective spectra.Comment: 67 pages, minor changes, accepted versio
The special fiber of the motivic deformation of the stable homotopy category is algebraic
For each prime , we define a -structure on the category
of harmonic
-motivic left module spectra over , whose
MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent
to the abelian category of -completed -comodules that are
concentrated in even degrees. We prove that
is equivalent to
as stable
-categories equipped with -structures.
As an application, for each prime , we prove that the motivic Adams
spectral sequence for , which converges to the motivic
homotopy groups of , is isomorphic to the algebraic
Novikov spectral sequence, which converges to the classical Adams-Novikov
-page for the sphere spectrum . This isomorphism of
spectral sequences allows Isaksen and the second and third authors to compute
the stable homotopy groups of spheres at least to the 90-stem, with ongoing
computations into even higher dimensions.Comment: Accepted version, 85 page
The reduced ring of the -graded -equivariant stable stems
We describe in terms of generators and relations the ring structure of the
-graded -equivariant stable stems modulo the
ideal of all nilpotent elements. As a consequence, we also record the ring
structure of the homotopy groups of the rational -equivariant sphere
.Comment: 13 page
Stable homotopy groups of spheres
We discuss the current state of knowledge of stable homotopy groups of
spheres. We describe a new computational method that yields a streamlined
computation of the first 61 stable homotopy groups, and gives new information
about the stable homotopy groups in dimensions 62 through 90. The method relies
more heavily on machine computations than previous methods, and is therefore
less prone to error. The main mathematical tool is the Adams spectral sequence