ON THE IMMERSION PROBLEM FOR REAL PROJECTIVE SPACES

Hence, and from the Hurwitz-Radon theorem, it follows that if P' can be immersed in (2j — k)-space, when j + 1 is a power of 2, then P admits a fc-field of linearly independent tangent vectors. Our method is based on the work of Adams [ l ] , Atiyah [2] and Hopf [6]. In [3] Atiyah has proved, by the method of exterior powers, that P' cannot be immersed in (2j —£)-space where p is approximately j / 2 . When j+1 is a power of 2 the value of p in all cases exceeds the value of q in (1.1), and p/q~2~/r when r is large. For r ^ 3 it is known (see Hirsch [5]) that P can be immersed in C/'+l)-space. I do not know whether P can be immersed in 22space, P in 53-space, and so forth.