Unresolved higher-degree unstable homotopy groups of unitary groups

Determine the unstable homotopy groups [?]_{2r+j}U(r), particularly their torsion, for values of j large relative to the prime, where existing results leave substantial cases unknown.

Background

The paper surveys classical computations of unstable homotopy groups [?]_{2r+j}U(r), emphasizing that many cases are known for small j and that partial results exist for prime-power torsion at fixed primes. The authors identify a remaining gap when the degree offset j is large compared with the prime, motivating their use of topological modular forms and higher real K-theories to detect new torsion families.