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 introduce the so-called doubling metric on the collection of non-empty bounded open subsets of a metric space. Given an open subset U of a metric space X, the predecessor U_∗ of U is defined by doubling the radii of all open balls contained inside U, and taking their union. The predecessor of U is an open set containing U. The directed doubling distance between U and another subset V is the number of times that the predecessor operation needs to be applied to U to obtain a set that contains V. Finally, the doubling distance between open sets U and V is the maximum of the directed distance between U and V and the directed distance between V and U.

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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.

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].