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Some remarks on the root invariant. (English) Zbl 0912.55011

Dwyer, William G. (ed.) et al., Stable and unstable homotopy. Proceedings of workshops held during 1996 at the Fields Institute, Waterloo, Canada. Providence, RI: American Mathematical Society. Fields Inst. Commun. 19, 31-38 (1998).
The root invariant (called the Mahowald invariant by some authors) assigns to each element \(x\neq 0\) in the stable homotopy of spheres a nonzero coset \(R(x)\) in a higher stem. It seems to be a fundamental concept in stable homotopy. In a previous paper, the author and J. Greenlees [Exp. Math. 4, No. 4, 289-297 (1995; Zbl 0858.55012)] gave a simple definition of \(R(x)\) based on equivariant homotopy. This definition is used here to derive a relation between \(R(xy)\) and \(R(x)R(y)\). The Adams spectral sequence leads to an algebraic companion to the root invariant. The second theorem of the present paper is concerned with this algebraic analogue. This result and the discussion which follows shed some light on the question in which band of stems \(R(x)\) can be lying.
For the entire collection see [Zbl 0890.00047].

MSC:

55Q45 Stable homotopy of spheres
55Q91 Equivariant homotopy groups
55P42 Stable homotopy theory, spectra
55P91 Equivariant homotopy theory in algebraic topology
55T15 Adams spectral sequences

Citations:

Zbl 0858.55012
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