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We prove that the 2-primary $\pi_{61}$ is zero. As a consequence, theKervaire invariant element $\theta_5$ is contained in the strictly defined4-fold Toda bracket $\langle 2, \theta_4, \theta_4, 2\rangle$.  Our result has a geometric corollary: the 61-sphere has a unique smoothstructure 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 ofthis paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove thisdifferential by introducing a new technique based on the algebraic andgeometric Kahn-Priddy theorems. The success of this technique suggests atheoretical way to prove Adams differentials in the sphere spectrum inductivelyby use of differentials in truncated projective spectra.

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