3-primary v1-periodic homotopy groups of E7

TL;DR

This paper computes the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7, filling a gap in the understanding of odd-primary homotopy groups of compact simple Lie groups.

Contribution

It provides the first computation of the 3-primary v1-periodic homotopy groups of E7, using the unstable Novikov spectral sequence and recent convergence results.

Findings

Computed the 3-primary v1-periodic homotopy groups of E7.

Established convergence of the spectral sequence to these groups.

Extended the understanding of odd-primary homotopy groups of exceptional Lie groups.

Abstract

We compute the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7. Now E8 at the primes 3 and 5 is the only compact simple Lie group whose odd-primary v1-periodic homotopy groups remian to be computed. The main work is computing the unstable Novikov spectral sequence of \Omega E7/Sp(2). Showing that this converges to v1-periodic homotopy groups requires recent work of Bousfield and Bendersky-Thompson.