On periodic families in the stable stems of height two

TL;DR

This paper identifies numerous infinite periodic families in the 2-primary stable homotopy groups of spheres, confirming predictions and revealing their detection in TMF fixed points, with implications for exotic spheres in specific dimensions.

Contribution

It uncovers new infinite periodic families in stable homotopy groups, confirms predictions by Hopkins--Mahowald, and links these families to TMF fixed points and exotic spheres.

Findings

Discovered infinite periodic families in stable homotopy groups.

Confirmed existence of families predicted by Hopkins--Mahowald.

Linked families to fixed points of TMF and exotic spheres in specific dimensions.

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.