Stable Stems

By Daniel Isaksen. Memoirs of the American Mathematical Society, 2014

83 citations recorded by the source corpus.

We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. We then use the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. In addition to finding all Adams differentials in this range, we also resolve all hidden extensions by 2, η, and ν, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences. We also compute the motivic stable homotopy groups of the cofiber of the motivic element τ . This computation is essential for resolving hidden extensions in the Adams spectral sequence. We show that the homotopy groups of the cofiber of τ are the same as the E2-page of the classical Adams-Novikov spectral sequence. This allows us to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8a8c46f99cc1ca7524f323bee9ddca45469eb279

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Citations in the corpus

Low-dimensional Milnor–Witt stems over ℝ (2015, methodology): “The groups ExtC have been computed in [Dugger and Isaksen 2010; Isaksen 2014b] through a large range of dimensions.”
The special fiber of the motivic deformation of the stable homotopy category is algebraic (2018, background): “Statement (2) is proved by Isaksen in Table 42 of stable stems [29].”
Topological resolutions in K(2) ‐local homotopy theory at the prime 2 (2016, background): “, see the chart \The E 1-page of the classical Adams Spectral Sequence" in [Isa14]. The fact that some class detected by h2 3h5 can have a non-zero multiplication can be found in Lemma 4.114 of [Isa]. See also Table 33 of that paper. Note that Isaksen is careful to label this lemma as tentative”