The Adams-Novikov E_2-term for Behrens' spectrum Q(2) at the prime 3

TL;DR

This paper computes the Adams-Novikov E_2-term for Behrens' spectrum Q(2) at prime 3, revealing its potential to detect the algebraic beta family in the 3-local sphere's homotopy groups.

Contribution

It provides the first detailed computation of the Adams-Novikov E_2-term for Q(2) at prime 3, linking it to the detection of the algebraic beta family.

Findings

The E_2-term computation supports the conjecture about detecting the beta family.

Techniques from rational homotopy are effectively applied to this spectral computation.

The work suggests Q(2) captures key elements of the 3-primary stable homotopy groups.

Abstract

We compute the Adams-Novikov E_2-term of a spectrum Q(2) constructed by Behrens. The homotopy groups of Q(2) are closely tied to the 3-primary stable homotopy groups of spheres; in particular, they are conjectured to detect the homotopy beta family of Greek letter elements at the prime 3. Our computation leverages techniques used by Behrens to compute the rational homotopy of Q(2), and leads to a conjecture that the Adams-Novikov E_2-term for Q(2) detects the algebraic beta family in the BP-based Adams-Novikov E_2-term for the 3-local sphere.