Vector bundles over \((8k+3)\)-dimensional manifolds (Q797876)

Let M be a closed, connected, smooth and 3-connected mod 2 (i.e. \(H_ i(M;{\mathbb{Z}}_ 2)=0\), \(0<i\leq 3)\) manifold of dimension \(3+8k\) with \(k>1\). We obtain some necessary and sufficient condition for the span of a \((3+8k)\)-plane bundle \(\xi\) over M to be greater than or equal to 7 or 8. We obtain, for M 4-connected mod 2 and satisfying \(Sq^ 2Sq^ 1H^{n-8}(M;{\mathbb{Z}}_ 2)=Sq^ 2H^{n-7}(M;{\mathbb{Z}}_ 2),\) where \(n=\dim\) \(M\equiv 11 mod 16\) with \(n>11\), that span \(M\geq 8\) if and only if the Kervaire mod 2 semi-characteristic \(\kappa_ 2(M)\) of M is zero. Some applications to product manifolds and immersions are given.