4-fields on \((4k+2)\)-dimensional manifolds (Q1079238)

Let M be a closed, connected, smooth and 2-connected mod 2 (i.e., \(H_ i(M;{\mathbb{Z}}_ 2)=0\), \(0<i\leq 2)\) manifold of dimension \(n=4k+2\) with \(k>1\). We obtain some necessary and sufficient conditions for the span of an n-plane bundle \(\eta\) over M to be greater than or equal to 4. For instance for k odd, span \(M\geq 4\) if, and only if \(X(M)=0\). Some applications to immersions are given. In particular if \(n=2+2^{\ell}\), \(\ell \geq 3\) and \(w_ 4(M)=0\) then M immerses in \({\mathbb{R}}^{2n-4}\).