Towards a splitting of the $K(2)$-local string bordism spectrum

TL;DR

This paper demonstrates a splitting of the $K(2)$-local string bordism spectrum into Morava $E$-theories, revealing new structural insights in chromatic homotopy theory.

Contribution

It introduces a novel splitting of the $K(2)$-local string bordism spectrum involving the spectrum $T_2$ and Morava $E$-theories, expanding understanding of $K(2)$-local spectra.

Findings

The $K(2)$-local string bordism spectrum splits into Morava $E$-theories.

The spectrum $T_2$ is related to the Thom spectrum over $ ext{Ω}SU(4)$.

The splitting provides new tools for studying chromatic phenomena at height 2.

Abstract

We show that $K(2)$-locally, the smash product of the string bordism spectrum and the spectrum $T_2$ splits into copies of Morava $E$-theories. Here, $T_2$ is related to the Thom spectrum of the canonical bundle over $\Omega SU(4)$.