Block bundle obstruction to Kervaire invariant one

PL block bundle theory provides obstructions to the existence of Kervaire invariant one elements theta(j) in the stable stems pi(s)(k) for k = 2(j+1) -2. A fundamental theorem of Barratt-Jones-Mahowald with extensions by Minami and Knapp provides a lower bound for the sphere-of-origin of Kervaire invariant one elements. This lower bound is the sphere-of-origin for theta(4), theta(5), and also theta(6) if it exists. The existence of theta(j) is equivalent to the vanishing of a certain Whitehead product involving the Im J generator beta(j) for 4 <= j <= 6.