nLab Arf-Kervaire invariant problem

Idea

The Arf-Kervaire invariant problem asks whether certain hypothetical elements $\theta_j \in \pi^s_{2^{j+1} - 2}$ 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 $d\neq 2^{j+1} - 2$.

Prior to (Hill-Hopkins-Ravenel 09), all that was known rested on the explicit construction of such elements for $j=1,...,5$ (so in dimensions 2,6,14,30 and 62). HHR established that these elements do not exist for $j \gt 6$, so the only dimension in which existence remains unknown in 126 (ie $j=6$). The proof is by construction of a 256-periodic spectrum $\Omega$ and a spectral sequence for it that can detect the elements $\theta_j$ as elements of $\pi_*(\Omega)$. HHR then show that $\pi_n(\Omega)=0$ for $-4\lt n\lt 0$, which by the periodicity, implies that the images of $\theta_j$ must be elements of the trivial group, and hence are themselves trivial.

Key properties of the $C_8$ fixed point spectrum $\Xi$

We write $\Xi$ to mean the spectrum “$\Omega$” discussed in Hill-Hopkins-Ravenel 09.

Detection theorem

It has an Adams-Novikov spectral sequence in which the image of each $\theta_j$ is non-trivial. This means if $\theta_j\in \pi_*(\mathbb{S})$ exists then it can be seen in $\pi_*(\Xi)$.

Periodicity Theorem

The spectrum is 256-periodic, as in $\Omega^{256}\Xi \simeq \Xi$.

Gap Theorem

We have $\pi_k(\Xi) = 0$ for $-4 \lt k \lt 0$. Its proof uses the slice spectral sequence.

Result

Suppose $\theta_7 \in \pi_{254}(\mathbb{S})$ exists then the detection theorem implies that it has a non-trivial image in $\pi_{254}(\Xi)$. But by the periodicity and gap theorems we see that $\pi_{254}(\Xi)$ is trivial. The argument for $j \ge 7$ is similar since $|\theta_j| = 2^{j+1} \equiv -2 \mod 256$.

Related concepts

• Kervaire invariant

• Hopf invariant one problem

References

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:

• Hill-Hopkins-Ravenel 09, Appendix

• HillHopkinsRavenel, section 4

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)