The triviality of the 61-stem in the stable homotopy groups of spheres. (English) Zbl 1376.55013

Wang, Guozhen; Xu, Zhouli

Ann. Math. (2) 186, No. 2, 501-580 (2017).

Let \(\theta_n\in \pi_{2^{n+2}-2}\) denote the Kervaire invariant one element where we know that \(\theta_1\), \(\theta_2\) and \(\theta_3\) can be chosen to be \(\eta^2\), \(\nu^2\) and \(\sigma^2\) respectively, but it does not exist for all \(n \geq 7\). Moreover, it is known that both \(\theta_4\) and \(\theta_5\) exist as elements such that \(2\theta_4=2\theta_5=0\) and \(\theta_4^2=\theta_5^2=0\). The purpose of this paper is to prove the following: (i) The 2-primary component of \(\pi_{61}\) is zero; (ii) \(\theta_5\) is contained in the strictly defined 4-fold Toda bracket \(\langle 2, \theta_4, \theta_4, 2 \rangle\); (iii) \(S^{61}\) has a unique smooth structure. The last two assertions (ii) and (iii) are proved in the first section, assuming that (i) holds true, which is roughly sketched below. The proof of (i) is given in Section 2 or later. Since \(\langle 2, \theta_4, \theta_4 \rangle \subset {}_2\pi_{61}\), we have \(\langle 2, \theta_4, \theta_4 \rangle=0\) due to (i) so we see that \(\langle 2, \theta_4, \theta_4, 2 \rangle\) is strictly defined. If \(h_n \in \text{Ext}_2^{1, 2^n}\) is an element in the Adams spectral sequence (ASS) for spheres such that its square \(h_n^2\) detects \(\theta_n\), then because of \(d_2(h_5)=h_0h_4^2\) it follows that \(\langle h_0, h_4, h_4, h_0 \rangle=h_5^2\) holds in the \(E_3\)-page. From this (ii) follows immediately. Let \(\Theta_n\) be the group of \(h\)-cobordism classes of homotopy \(n\)-spheres and \(\Theta_n^{bp}\) the subgroup defined by homotopy spheres which bound parallelizable manifolds. Then from the fact that \(\theta_5\) is a Kervaire invariant one element we have that \(\Theta_{61}^{bp}=0\), so that \(\Theta_{61}=\pi_{61}/\text{Im} J\). This implies that when shown that \(\text{Coker} J\) in dimension 61 vanishes at all odd primes, we obtain \(\Theta_{61}=0\) by combining this fact with (i) and thus we have (iii).
The proof of (i) essentially follows the computation of \(\pi_*\) by D. C. Isaksen [“Classical and motivic Adams charts”, Preprint, arXiv:1401.4983], developing a new method which allows us to overcome the lack of computability of differentials in the ASS for the 2-local sphere. In fact, we have three elements in the \(E_\infty\)-page whose differentials are unknown. But by determining their respective differentials based on this new method, the authors succeed in proving that there are no elements left in the \(E_\infty\)-page of the 61-stem. In terms of the notation used therein, they are described as follows: \[ d_3(D_3)=B_3, \;d_5(A')=h_1B_{21} \;\text{and} \;d_4(h_1X_1)=gz. \] Here the last two relations are obtained, respectively, in Sections 12 and 11. However, the proof of the first relation requires a lengthy technical argument in Sections 3-10.

Reviewer: Haruo Minami (Nara)

MSC:

55Q45

Stable homotopy of spheres

Keywords:

stable homotopy groups of spheres; smooth structures; Adams spectral sequence; Atiyah-Hirzebruch spectral sequence; cell diagrams

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