\(\mathrm{tmf}\)-based Mahowald invariants
The Greek letter elements were defined in order to organize the stable homotopy groups of spheres into periodic families of elements of longer an longer wavelength, corrsponding to higher and higher chromatic heights. One way to construct Greek letter elements at the prime \(2\) is to take iterated Mahowald invariants of the powers of \(2\) in degree zero in the stable homotopy groups of spheres. This has the advantage that these homotopy Greek letter elements are always nontrivial. For example, the \(2\)-primary homotopy beta family \(\beta_{i}^{h}\) is the Mahowald invariant of the Mahowald invariant of \(2^{i}\). Such Greek letter elements are well understood at height one, but at higher heights they are only known explicitly in a range at the prime \(2\). One way to approximate Mahowald invariants is to use generalized homology theories such as real K-theory and topological modular forms. This reduces a largely intractable problem, depending on computations of the entire stable homotopy groups of spheres, so a significantly more tractable problem depending on the coefficients of your generalied homology theory. For example, the paper under review presents clean and detailed computations of iterated Mahowald invariants using topological modular forms. This gives an approximation to the iterated Mahowald invariants for the sphere spectrum and in fact recovers iterated Mahowald invaraints for the sphere in some low dimensions. The paper is elegant and well written and it includes numerous figures that cleanly illustrate the computations.
scientific article; zbMATH DE number 7611768
scientific article