Loop-space status of the metastable 7-sphere

Determine whether the metastable sphere \(S^7\) is a loop space.

Background

The paper uses injectivity constraints for suspension maps and the classical Hopf invariant one theorem to rule out loop-space structures for metastable spheres other than dimensions 0, 1, 3, and 7. Direct computations then exclude the first three remaining cases.

The case $S^7$ lies beyond the range of the computations presented, so the paper leaves unresolved whether this final candidate is a loop space.