For any element in an odd stem on a sphere of odd dimension, there is a space B realized as the total space of a (non-split) spherical fibration such that its localization at primes \(p \geq 5\) is an H-space, but at the prime 3 there are obstructions. The new result is that the space B built from any element in Toda’s 3-primary alpha sequence on the 3-sphere is a 3-local H-space.

There will be a pretalk at 2 in 1106A concerning the finite homotopy types that can support an H-structure.