On the $\eta$-inverted sphere

TL;DR

This paper proves that the first two homotopy groups of the η-inverted sphere spectrum vanish over fields of characteristic not two and provides a cell presentation of higher Witt theory over complex numbers.

Contribution

It establishes the vanishing of initial homotopy groups of the η-inverted sphere spectrum and offers a cell presentation of higher Witt theory over complex numbers.

Findings

First and second homotopy groups are zero over certain fields.

Provides a cell presentation of higher Witt theory over complex numbers.

Abstract

It is shown that the first and second homotopy groups of the $\eta$-inverted sphere spectrum over a field of characteristic not two are zero. A cell presentation of higher Witt theory is given as well, at least over the complex numbers.