Rationalized stable homotopy groups of spheres

Computing the stable homotopy groups of spheres \(\pi_*^s\) is a major problem in stable homotopy theory.

In chromatic homotopy theory one studies, for every prime \(p\), a tower \[\cdots \to L_n\mathcal{S} \to \cdots \to L_1\mathcal{S} \to \mathcal{S}_\mathbb{Q}\] over the rational sphere spectrum \(\mathcal{S}_\mathbb{Q}\) whose homotopy limit is the \(p\)-localization \(\mathcal{S}_{(p)}\) of the sphere spectrum. Since \(\pi_*(\mathcal{S}_{(p)})\), if known for every prime \(p\), determines \(\pi_*(\mathcal{S}) = \pi_*^s\), one has “reduced” by this to the problem of computing \(\pi_*(L_n \mathcal{S})\).

Now for the following we consider the \(K(n)\)-local spheres \(L_{K(n)}\mathcal{S}\) which are, in a certain sense, the difference between \(L_n \mathcal{S}\) and \(L_{n-1} \mathcal{S}\).

In a recent preprint by Barthel-Schlank-Stapleton-Weinstein (arXiv:2402.00960), which one might probably call a major breakthrough if it turns out to be correct, a complete computation of the rationalizations of \(\pi_*(L_{K(n)} \mathcal{S})\) was achieved: There is an isomorphism of graded \(\mathbb{Q}\)-algebras \[\mathbb{Q} \otimes \pi_*(L_{K(n)} \mathcal{S}) \cong \Lambda_{\mathbb{Q}_p}(\zeta_1, \ldots, \zeta_n),\] where the latter is the exterior \(\mathbb{Q}_p\)-algebra on generators \(\zeta_i\) in degree \(1-2i\).

edit: Thanks to Tobias Barthel, who pointed out a mistake in an earlier version of this post.

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