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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.

We establish a differential d 2 (D 1)= h 0 2 h 3 g 2 in the 51-stem of the Adams spectral sequence at the prime 2, which gives the first correct calculation of the stable 51 and 52 stems. This differential is remarkable since we know of no way to prove it without recourse to the motivic Adams spectral sequence. It is the last undetermined differential in the range of the first author's detailed calculations of the n-stems for n< 60 [6]. This note advertises the use of the motivic Adams spectral sequence to obtain information about classical stable homotopy groups.

We show that the Hopf elements, the Kervaire classes, and the -family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the C 2-fixed points of the Real bordism spectrum. A subset of these families is detected by the C 2-fixed points of Real JohnsonWilson theory E R (n), depending on n. In the proof, we establish an isomorphism between the slice spectral sequence and the C 2-equivariant May spectral sequence of B P R.

The E_1-term of the (2-local) bo-based Adams spectral sequence for the sphere spectrum decomposes into a direct sum of a v_1-periodic part, and a v_1-torsion part. Lellmann and Mahowald completely computed the d_1-differential on the v_1-periodic part, and the corresponding contribution to the E_2-term. The v_1-torsion part is harder to handle, but with the aid of a computer it was computed through the 20-stem by Davis. Such computer computations are limited by the exponential growth of v_1-torsion in the E_1-term. In this paper, we introduce a new method for computing the contribution of the v_1-torsion part to the E_2-term, whose input is the cohomology of the Steenrod algebra. We demonstrate the efficacy of our technique by computing the bo-Adams spectral sequence beyond the 40-stem.

In this note, we use Curtis's algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of \mathbb {R} P^\infty with the aid of a computer, which gives us its Adams \mathbb {R} P^\infty -page in the range of \mathbb {R} P^\infty . We also compute the transfer map on the Adams \mathbb {R} P^\infty -pages. These data are used in our computations of the stable homotopy groups of \mathbb {R} P^\infty in [6] and of the stable homotopy groups of spheres in [7].

We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2primary part of 5 1 is 8 8 2. This was the last unsolved 2extension problem left by the recent work of Isaksen and the authors through the 6 1stem.

We prove that the 61-sphere has a unique smooth structure. Following results of Moise [35], Kervaire-Milnor [25], Browder [10] and Hill-Hopkins-Ravenel [19], we show that the only odd dimensional spheres with a unique smooth structure are S1, S3, S5 and S61.

Among all the interesting spaces in topology, the spheres are no doubt beautiful objects and of most consideration since antiquity. Any nontrivial observations of them are of course very important. From the categorical point of view, we should not just focus on the objects themselves, but the morphisms between them as well. For this purpose, in algebraic topology, we do want to classify the set of continuous morphisms between spheres under the equivalent relation named homotopy, which describes a continuous deformation between two continuous maps. Let Sn be the n-sphere and k (X) be the set of homotopic equivalent based maps from Sk to X. For the reason that Sk is a double suspension when k 2, the set is actually an abelian group. A natural question is, what are these abelian groups?

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