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?

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.

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.

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].

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.