The extended power construction can be used to create new framed manifolds out of old. We show here how to compute the effect of such operations in the Adams spectral sequence, extending partial results of Milgram and the author. This gives the simplest method of proving that Jones’ 30-manifold has Kervaire invariant one, and allows the construction of manifolds representing Mahowald’s classes η4 and η5, among others.
Algebraic Geometry | Geometry and Topology
Bruner, R. R. Extended powers of manifolds and the Adams spectral sequence. In J.P.C. Greenlee, R.R. Bruner, & N. Kuhn, Eds. Homotopy methods in algebraic topology. Contemp. Math. 271, 41-51 (2001). Providence, RI: American Mathematical Society.