nLab
Arf-Kervaire invariant problem

Contents

Contents

Idea

The Arf-Kervaire invariant problem asks whether certain hypothetical elements θ jπ 2 j+12 s\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 d2 j+12d\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,...,5j=1,...,5 (so in dimensions 2,6,14,30 and 62). HHR established that these elements do not exist for j>6j \gt 6, so the only dimension in which existence remains unknown in 126 (ie j=6j=6). The proof is by construction of a 256-periodic spectrum Ω\Omega and a spectral sequence for it that can detect the elements θ j\theta_j as elements of π *(Ω)\pi_*(\Omega). HHR then show that π n(Ω)=0\pi_n(\Omega)=0 for 4<n<0-4\lt n\lt 0, which by the periodicity, implies that the images of θ j\theta_j must be elements of the trivial group, and hence are themselves trivial.

Key properties of the C 8C_8 fixed point spectrum Ξ\Xi

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

Detection theorem

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

Periodicity Theorem

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

Gap Theorem

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

Result

Suppose θ 7π 254(𝕊)\theta_7 \in \pi_{254}(\mathbb{S}) exists then the detection theorem implies that it has a non-trivial image in π 254(Ξ)\pi_{254}(\Xi). But by the periodicity and gap theorems we see that π 254(Ξ)\pi_{254}(\Xi) is trivial. The argument for j7j \ge 7 is similar since |θ j|=2 j+12mod256|\theta_j| = 2^{j+1} \equiv -2 \mod 256.

References

The solution of the problem in the negative, except for one outstanding dimension (namely 126), appears in

On the equivariant stable homotopy theory involved:

More resources are collected at

Last revised on March 24, 2019 at 19:20:45. See the history of this page for a list of all contributions to it.