Hurewicz Images of Real Bordism Theory and Real Johnson-Wilson Theories

TL;DR

This paper demonstrates how key elements in the stable homotopy groups of spheres are detected via the Hurewicz map using Real Brown-Peterson and Johnson-Wilson spectra, revealing new detection methods in equivariant homotopy theory.

Contribution

It establishes the detection of important homotopy elements by the Hurewicz map through $C_2$-fixed points of Real spectra, connecting classical and equivariant stable homotopy theory.

Findings

Detection of Hopf elements, Kervaire classes, and $ar{ ext{kappa}}$-family by the Hurewicz map.

Identification of certain families detected by Real Johnson-Wilson theories.

Extension of classical detection results to equivariant and Real spectra.

Abstract

We show that the Hopf elements, the Kervaire classes, and the $\bar{\kappa}$-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the $C_2$-fixed points of the Real Brown-Peterson spectrum. A subset of these families is detected by the $C_2$-fixed points of Real Johnson-Wilson theory $E\mathbb{R}(n)$, depending on $n$.