The root invariant in homotopy theory
The root invariant (also known as the Mahowald invariant) of an element in the 2-primary stable homotopy groups of spheres is a coset of elements defined as follows. Represent the class \(\alpha\) by a map \(S^{t - 1} \to S^{-1}\), and choose the smallest \(n\) such that the composite \(S^{t - 1} \to S^{-1} \to P_{-n}\) is essential. Such an \(n\) exists by Lin's Theorem. This composite can be factored through a map \(g : S^{t-1} \to S^{-n}\), where \(S^{-n}\) is the bottom cell of \(P_{- n}\). The coset consisting of all such maps \(g\) is the root invariant \(R(\alpha)\). There is an odd-primary analogue of this, and other generalizations are presented in this paper. This paper presents a number of results about root invariants. One conjecture is that the root invariant of a \(v_ n\)-periodic element is \(v_{n + 1}\)-periodic. This paper offers evidence for this by proving that if \(p \geq 5\), then \(R(\alpha_ i) = \beta_ i\), where \(\alpha_ i\) (resp. \(\beta_ i\)) are standard \(v_ 1\)-periodic (resp. \(v_ 2\)- periodic) classes. One of the main results is that an element is anomalous if and only if it is a root invariant. The definition of an anomalous element involves the EHP spectral sequence. Any stable homotopy class gives rise to elements in the \(E_ 1\)-term of this spectral sequence on all sufficiently large lines. It is anomalous if for all \(r\), these elements are nonzero in \(E_ r\) on infinitely many lines. The authors conjecture that root invariants are quite rare. A conjecture is made about their asymptotic relative frequency. Many other topics are mentioned in this paper. One is a shortening of James periodicity in the EHP spectral sequence. Another deals with properties of Thom spectra indexed by the 2-adic integers which generalize stunted real projective spaces. The 2-adic integers arise as \(KO^*(RP^ \infty)\).
scientific article; zbMATH DE number 550348