The root invariant and \(v_ 1\)-periodic families
In this paper the author studies \textit{M. E. Mahowald} and \textit{D. C. Ravenel's} conjecture [Proc. Lefschetz Centen. Conf., Contemp. Math. 58.2, 57-74 (1984; Zbl 0639.55006)] that the root invariants of \(v_ i\)- periodic elements of \(\pi^ S_ *\) contain \(v_{i+1}\)-periodic elements. Let \(p\) be an odd prime and \(q=2p-2\). Suppose \(x\in\pi^ S_ *\) and \(f\in R(x)\). He proves: (i) If \(| R(x)|-| x|\equiv -1\pmod q\), then \(\langle f,p,\alpha_ k\rangle\cap R(p^ kx)\neq \emptyset\) or \(R(p^ kx)\) is in a higher dimension than \(\langle f,p,\alpha_ k\rangle\). (ii) If \(| R(x)| - | x| \equiv 0({}\bmod q)\), then \(\alpha_ kf\in R(p^ kx)\), or \(R(p^ kx)\) is in a higher dimension than \(\alpha_ k f\). For the root invariants of the degree-\(p^ k\)-map on the sphere, \(\alpha_ k\in R(p^ k\iota)\) hold for \(p>2\). The plan is as follows. First, the Brown-Peterson homology of \(B\mathbb{Z}/p\) and stunted \(B\Sigma_ p\) are calculated. Next, some low- dimensional groups \([X,Y]_ k\), where \(X\) and \(Y\) are either spheres, Moore spaces or \(V(1)\), are calculated by using known values of \(\pi^ S_ *\). These results are used for an obstruction-theory argument. Moreover some Ext-calculations, especially filtration 1-piece \(\text{Ext}^{1,*}_{BP_ *BP}(BP_ *,BP_ *(B\Sigma_ p))\), are described. This calculation is used to detect the root invariants of \(\alpha_ i\) for \(p>3\).
scientific article; zbMATH DE number 54632
scientific article