Why every smooth orientable 4-dimensional manifold admits an immersion into $\mathbb{R}^{6}$?
This is a one-line question as I see the statement in a comment by Michael Hopkins(update: this is wrong, it is by Peter Kronheimer) . I thought about it for a long time but I do not know how to prove it or to approach it. Characteristic classes provide a way of showing "if this...", but does not help to show the existence of such an immersion. (comment: this is stupid line of thinking because obviously I did not make use of all characteristic tools available to me, as evident in reading Kirby's book)
update: The correct statement is provided in Kirby's book, page 44 Lemma 1, which states such immersion exists iff there exists a characteristic class $x\in H^{2}(M^{4};\mathbb{Z})$ such that $x_{(2)}=-w_{2}$ and $x^{2}=-p_{1}$.