A note on immersing manifolds in euclidean spaces

Let M be a closed, connected smooth and 3-connected mod 2 (that is Hi(M;ℤ2) = 0, 0 < i ≤ 3) manifold of dimension n = 7 + 8k. Using a combination of cohomology operations on certain cohomology classes of M and on the Thom class of the stable normal bundle of M we show that under certain conditions M immerses in R2n−8. This extends previously known results for such a general manifold when the number of 1's in the dyadic expansion of n is less than 8.