On periodic families in the stable stems of height two

Abstract

We discover a host of infinite periodic families in the 2-primary stable homotopy groups of spheres. We also confirm the existence of many families predicted by Hopkins--Mahowald. These families appear in nineteen different congruence classes of degrees modulo 192, seven of them consist of simple 4-torsion elements, and another four of simple 8-torsion. They all vanish in the homotopy groups of the spectrum TMF of topological modular forms, but we show that they are detected in the fixed-points of TMF with respect to an Atkin--Lehner involution. As a consequence, we confirm the existence of exotic spheres in all dimensions congruent to 72, 144, and 168 modulo 192.