Extended powers of manifolds and the Adams spectral sequence (Q2752868)

Let \(D_rR\) denote the extended power construction of an \(H_\infty\)-Ring spectrum \(R\). There are compatible maps \(D_r(R)\rightarrow R\) extending the \(r\)-fold product map and this construction leads to operations in the homotopy groups of \(R\) and in the Adams spectral sequence convergent to the homotopy of \(R\). For \(R = S\), the sphere spectrum, the homotopy groups can be interpreted as bordism classes of framed manifolds, and the operations can be described geometrically. The paper under review describes the relationship among these constructions and, using calculations in the Adams spectral sequence, deduces some geometric results. It is shown that the natural framing on the Jones 30-manifold [\textit{J. D. S. Jones}, Topology 17, 249-266 (1978; Zbl 0413.55009)] has Kervaire invariant \(1\) (previously it was only known that some framing of this manifold had Kervaire invariant \(1\)) and a construction is given for manifolds representing the Mahowald elements \(\eta_4\) and \(\eta_6\).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].