New simple $η$-torsion families of elements in the stable stems

TL;DR

The paper identifies two new infinite η-torsion families in the stable stems at 2-local, 192-periodic scale by bootstrapping tmf-based information from $A_1$ through re-ordered cofiber sequences. The approach relies on the tmf-Hurewicz homomorphism to lift $tmf_*A_1$ classes to π_*S and filters to isolate simple $v_1$-torsion elements with vanishing tmf-Hurewicz image. The two families appear in dimensions $73+192k$ and $120+192k$ (for $k\in\mathbb{N}$), with the $k=0$ cases aligning with Adams-detection patterns and supported by diagrammatic and boundary-method arguments. These results also imply the existence of very exotic spheres in at least 102 congruence classes modulo 192 and indicate that these η-torsion elements survive in $T(2)$- and $K(2)$-local stable stems, highlighting robust connections between tmf methods and classical homotopy theory.

Abstract

We produce two new $192$-periodic infinite families of simple $η$-torsion elements in the stable homotopy groups of spheres using the $\mathit{tmf}$-Hurewicz homomorphism and the complex projective plane.

Theorems & Definitions (19)

• Theorem A
• Remark 1.1
• Remark 1.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof