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The Arf-Kervaire problem in algebraic topology: sketch of the proof. (English) Zbl 1249.55005
Jerison, David (ed.) et al., Current developments in mathematics, 2010. Somerville, MA: International Press (ISBN 978-1-57146-228-2/hbk). 1-43 (2011).
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”, 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.
The 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: \[ \pi^S_*(\Omega)=\pi^S_{*+256}(\Omega) \quad \text{and} \quad \pi^S_{-2}(\Omega)=0. \] 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 that if \(\theta_j \in \pi^S_{2^{j+1}-2}(S^0)\) is non-zero, then \(\iota_*(\theta_j)\neq 0\). Combining these two results implies that \(\theta_j=0\) for \(j \geq 7\).
Herein, the authors focus on the construction of \(\Omega\) and the proof of its properties. For the proof of the latter fact the reader is only referred to the main paper above.
For the entire collection see [Zbl 1245.00031].

55Q45 Stable homotopy of spheres
55P91 Equivariant homotopy theory in algebraic topology
57R55 Differentiable structures in differential topology
57R77 Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism)