The Arf-Kervaire invariant problem asks whether certain hypothetical elements of the stable homotopy groups of spheres exist. In fact the definition of these elements works in all dimensions but due to a theorem of Browder they do not exist in dimensions .
Prior to (Hill-Hopkins-Ravenel 09), all that was known rested on the explicit construction of such elements for (so in dimensions 2,6,14,30 and 62). HHR established that these elements do not exist for , so the only dimension in which existence remains unknown in 126 (ie ). The proof is by construction of a 256-periodic spectrum and a spectral sequence for it that can detect the elements as elements of . HHR then show that for , which by the periodicity, implies that the images of must be elements of the trivial group, and hence are themselves trivial.
We write to mean the spectrum “” discussed in Hill-Hopkins-Ravenel 09.
It has an Adams-Novikov spectral sequence in which the image of each is non-trivial. This means if exists then it can be seen in .
The spectrum is 256-periodic, as in .
We have for . Its proof uses the slice spectral sequence.
Suppose exists then the detection theorem implies that it has a non-trivial image in . But by the periodicity and gap theorems we see that is trivial. The argument for is similar since .
A solution of the problem in the negative, except for one outstanding dimension (namely 126), using methods of equivariant stable homotopy theory:
Michael Hill, Michael Hopkins, Douglas Ravenel, On the non-existence of elements of Kervaire invariant one, Annals of Mathematics 184 1 (2016)[doi:10.4007/annals.2016.184.1.1, arXiv:0908.3724, talk slides]
Michael Hill, Michael Hopkins, Douglas Ravenel, Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem, New Mathematical Monographs, Cambridge University Press (2021) [doi:10.1017/9781108917278]
Michael Hill, Michael Hopkins, Douglas Ravenel, The Arf-Kervaire problem in algebraic topology: Sketch of the proof, Current Developments in Mathematics, 2010: 1-44 (2011) (pdf, doi:10.4310/CDM.2010.v2010.n1.a1)
On the equivariant stable homotopy theory involved:
More resources are collected at
Douglas Ravenel, A solution to the Arf-Kervaire invariant problem, web resources 2009
Michael Hill, Michael Hopkins, Douglas Ravenel, The Arf-Kervaire invariant problem in algebraic topology: introduction (2016) (pdf)
Last revised on November 17, 2022 at 05:08:30. See the history of this page for a list of all contributions to it.