The triviality of the 61-stem in the stable homotopy groups of spheres

Guozhen Wang; Zhouli Xu

doi:https://doi.org/10.4007/annals.2017.186.2.3

Abstract

We prove that the 2-primary $\pi_{61}$ is zero. As a consequence, the Kervaire invariant element $\theta_5$ is contained in the strictly defined 4-fold Toda bracket $\langle 2, \theta_4, \theta_4, 2\rangle$.
Our result has a geometric corollary: the 61-sphere has a unique smooth structure, and it is the last odd dimensional case — the only ones are $S^1, S^3, S^5$ and $S^{61}$. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.

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@ARTICLE{Rok, author = {Rohlin, V. A.}, title = {On a mapping of the {$(n+3)$}-dimensional sphere into the {$n$}-dimensional sphere}, journal = {Doklady Akad. Nauk SSSR (N.S.)}, volume = {80}, year = {1951}, pages = {541--544}, mrclass = {56.0X}, mrnumber = {0046042}, mrreviewer = {P. J. Hilton}, zblnumber = {}, }
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@incollection{Tan2, author = {Tangora, Martin}, title = {Some extension questions in the {A}dams spectral sequence}, booktitle = {Proceedings of the {A}dvanced {S}tudy {I}nstitute on the {A}lgebraic {T}opology, {V}ol.{III}}, VENUE={{A}arhus {U}niv., {A}arhus, 1970}, pages = {578--587}, SERIES={Various Publ. Ser.}, VOLUME={13}, publisher = {Mat. Inst., Aarhus Univ., Aarhus}, year = {1970}, mrclass = {55E45}, mrnumber = {0339163}, mrreviewer = {R. M. F. Moss}, zblnumber = {0225.55018}, }
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@INCOLLECTION{Tan4, author = {Tangora, Martin}, title = {Some homotopy groups mod {$3$}}, booktitle = {Conference on Homotopy Theory}, venue = {{E}vanston, {I}ll., 1974}, series = {Notas Mat. Simpos.}, volume = {1}, pages = {227--245}, publisher = {Soc. Mat. Mexicana, México}, year = {1975}, mrclass = {55Q45}, mrnumber = {0761731}, zblnumber = {0334.55014}, }
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The triviality of the 61-stem in the stable homotopy groups of spheres

Abstract

Authors