The Arf-Kervaire problem in algebraic topology: sketch of the proof
type: Item
instance of: scholarly article
zbMATH Open document ID: 1249.55005
author: Douglas C. Ravenel
author: Michael A. Hill
author: Michael J. Hopkins
publication date: 2012-08-17T00:00:00Z
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 57R55
Mathematics Subject Classification ID: 57R77
zbMATH DE Number: 6066961
zbMATH Keywords: Arf-Kervaire invariant
zbMATH Keywords: framed manifolds
zbMATH Keywords: equivariant homotopy theory
zbMATH Keywords: Adams-Novikov spectral sequences
MaRDI profile type: Publication
title: The Arf-Kervaire problem in algebraic topology: sketch of the proof
review text: Let \(\theta_j\) denote the stable homotopy class in \(\pi^S_{2^{j+1}-2}(S^0)\) represented by a framed manifold of Arf-Kervaire invariant one. In the paper [``On the non-existence of elements of Kervaire invariant one'', \url{arXiv:0908.3724} (2009)] the authors prove that if \(\theta_j \in \pi_{2^{j+1}-2}\) is non-zero, then \(j \leq 6\), that is, stable homotopy classes with non-trivial Arf-Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62 and 126. This paper provides a detailed sketch of the proof of this result, which aims to introduce the reader to the techniques and strategy used therein.NEWLINENEWLINEThe approach taken to the proof can be outlined as follows. According to Browder's theorem, a framed \((2^{j+1}-2)\)-manifold with non-trivial Arf-Kervaire invariant one exists if and only if the element \(h_j^2\) in the \(E_2\)-term of the classical Adams spectral sequence is a permanent cycle (in the above expression \(\theta_j\) denotes an element detected by \(h_j^2\)). In the main paper mentioned above, the authors prove the non-existence of such a manifold by showing that this \(h_j^2\) cannot survive the Adams spectral sequence for \(j \geq 7\). But in fact, instead of using the classical Adams spectral sequence, the authors use the Adams-Novikov spectral sequence for the sphere, and show that the element of its \(E_2\)-term corresponding to each \(h_j^2\) with \(j \geq 7\) cannot be a permanent cycle. In order to prove this, the authors introduce a new multiplicative cohomology theory \(\Omega\) with the properties: NEWLINE\[NEWLINE\pi^S_*(\Omega)=\pi^S_{*+256}(\Omega) \quad \text{and} \quad \pi^S_{-2}(\Omega)=0.NEWLINE\]NEWLINE These properties tell us immediately that \(\pi^S_{2^{j+1}-2}(\Omega)=0\) for \(j \geq 7\). So by comparing the two Adams-Novikov spectral sequences for the sphere and \(\Omega\) via the unit map \(\iota : S^0 \to \Omega\), we can see that \(\iota_*(\theta_j)=0\) for \(j \geq 7\). On the other hand it is proved tha…
reviewed by: Haruo Minami
description: scientific article; zbMATH DE number 6066961