\(\mu\)-elements in \(S^ 1\)-transfer images
The \(\mu_ r\)-elements of Adams in the 2-primary component of the stable homotopy groups of spheres constitute a periodic family with rather extreme properties. Although these elements are in some sense very simple elements there are not known any framed manifolds representing them (except in the lowest dimensions). One possible method to find such manifolds is to write \(\mu_ r\) as an \(S^ 1\)- transfer image of some manifold with stable normal bundle isomorphic to a multiple of a complex line bundle. Here the \(S^ 1\)-transfer map \(t_ n\) is the boundary map belonging to the standard cofibre sequence \[ P_{m-n+1}\mathbb{C}^{(n- 1)\xi}\to P_{m-n}\mathbb{C}^{n\xi}@>t_ n>> S^{2n-1}\to \Sigma P_{m- n+1}\mathbb{C}^{(n-1)\xi} \] (for \(-\infty < n\leq m \leq \infty\)) of Thom spectra of multiples of the canonical complex line bundle on complex projective spaces. In this paper the problem for which \(n\), \(m\) the element \(\mu_ r\) is in \(\text{im}(t_{n*})\) is investigated. The main result is that for \(r\geq 0\) the numerical condition \({8r+2k+1\choose 4r+1}+2{8r+2k-1\choose 4r- 1}\not\equiv 0\bmod 4\) implies \(\mu_ r\in\text{im}(t_{2k*})\). For \(n\) odd \(\mu_ r\) is never in \(\text{im}(t_{n*})\). The \(\mu_ r\)-family is detected by the KO-theory Hurewicz map \(h_{KO}\) (=\(d\)-invariant). For the weaker problem whether \(\mu_ r+x\) for some \(x\) in \(\text{ker}(h_{KO})\) is in \(\text{im}(t_{n*})\), the author gives necessary and sufficient conditions in terms of the index of the Hurewicz map \(h: \pi^ S_{2m}(P_ \infty\mathbb{C}^{n\xi})\to H_{2m}(P_ \infty\mathbb{C}^{n\xi})\) and denominators of the power series \(((e^ x- 1)/x)^ j\). The corresponding problem for the \(S^ 0\)-transfer and real projective spaces is also treated. To prove his results the author uses the classical Adams spectral sequence and some known results on stable James numbers.
scientific article; zbMATH DE number 91632