Let \(\Theta_n\) be the \(n\)th homotopy sphere cobordism group under the connected sum operation in the sense of M. A. Kervaire and J. W. Milnor [Ann. Math. (2) 77, 504–537 (1963; Zbl 0115.40505)], and let \[ J : \pi_n SO \rightarrow \pi_n^s \] be the \(J\)-homomorphism, where \(\pi_*^s\) is the stable homotopy groups of spheres. Kervaire and Milnor proved that there is an isomorphism \[ \Theta_{4k} \cong\mathrm{coker} J_{4k} \] and there exist exact sequences \[ 0 \longrightarrow \Theta_{2k+1}^{bp} \longrightarrow \Theta_{2k+1} \longrightarrow \mathrm{coker} J_{2k+1} \longrightarrow 0 \] and \[ 0 \longrightarrow \Theta_{4k+2}\longrightarrow \mathrm{coker}~J_{4k+2} \overset{\Phi_K}\longrightarrow \mathbb{Z}/2 \longrightarrow\Theta_{4k+1}^{bp} \longrightarrow 0,\] where \(\Theta_{n}^{bp}\) is the subgroup of the group \(\Theta_n\) of homotopy \(n\)-spheres which bound a parallelizable manifold and \(\Phi_K\) is the Kervaire invariant; see also [J. Milnor, Notices Am. Math. Soc. 58, No. 6, 804–809 (2011; Zbl 1225.01040)]. W. Browder showed in [Ann. Math. (2) 90, 157–186 (1969; Zbl 0198.28501)] that the Kervaire invariant is zero for dimensions \(\neq 2^i -2\). For dimension \(2^i -2\), there is a framed manifold with Kervaire invariant one if and only if in the Adams spectral sequence for the stable homotopy groups of spheres the element \(h_i^2 \in E^2\) survives to \(E^\infty\). The work of M. A. Hill et al. [Ann. Math. (2) 184, No. 1, 1–262 (2016; Zbl 1366.55007)] is a remarkable paper in homotopy theory on the Kervaire invariant one problem except for in dimension \(126\). G. Wang and Z. Xu showed in [Ann. Math. (2) 186, No. 2, 501–580 (2017; Zbl 1376.55013)] that the \(61\)-stem in the stable homotopy groups of spheres is trivial.
In this beautiful paper in algebraic topology under review, the authors construct non-trivial elements of coker \(J\) in low degrees using Toda brackets of elements in the stable homotopy groups of spheres and chromatic homotopy theory to produce non-trivial \(v_2\)-periodic families. By using a modified Adams spectral sequence for \(M(8, v_1^8)\) and the computation of the \(E_1\)-page of the algebraic tmf resolution, they show that for every even \(k\) less than \(140\), coker\(J_k\) has a non-trivial element of Kervaire invariant \(0\), except for \(k = 2,4,6,12\) and \(56\). The remarkable results of this paper along with the works of Kervaire-Milnor, Hill-Hopkins-Ravenel and Wang-Xu assert that the only dimensions \(< 140\) for which \(S^n\) has a unique differentiable structure are \(1,2,3,5,6,12,56,61\) and perhaps \(4\). They also construct some non-trivial elements of \((\mathrm{coker}J_{4k})_{(p)}\) and \((\mathrm{coker}J_{8k-2})_{(p)}\) with Kervaire invariant \(0\) below dimension \(200\), where \(p = 2,3\) and \(5\).