The existence of 7-fields and 8-fields on \((8k+5)\)-dimensional manifolds (Q909295)

Let M be a cloed, connected and smooth manifold of dimension n congruent to 5 mod 8 and \(n\geq 21\). If, furthermore, M is 5-connected mod 2, the author offers necessary and sufficient criteria for the existence of 7- and 8-fields on M in terms of the Stiefel-Whitney class \(w_{n-7}(M)\), the Kervaire mod 2 semicharacteristic \(\chi_ 2(M)\) and a certain stable secondary cohomology operation. The proof involves detailed calculations in the modified Postnikov tower of an associated lifting problem. The methods also apply to yield certain results concerning immersions of M into \({\mathbb{R}}^{2n-8}\).