Nonimmersions of complex Grassmann manifolds

Abstract

If an oriented manifold M immerses in codimension k, then the normal bundle has dimension k such that its Euler class χ є Hk(M; Z) and χ2 є H2k(M; Z). (Cf. (3)).If M is the complex Grassmann manifold G2(Cn) of 2-planes in Cn (n = 4, 5,…, 15, 17), then dim M = 4n – 8 ≡ d and we shall show that although M immerses in R2d–1 by classical results (3), M does not immerse in Rd+d/2.The same result was obtained for n = 4 and 5 by Connell (2) and for n = 6 and 7 by the author (6). The nonimmersion results of this paper are new for n = 8, 9, …, 15, 17 and they are an improvement over the result for the general G2(Cn) obtained in (5). In this paper, we use generators of the cohomology ring of G2(Cn) different from those used in (2) and (6) and this simplifies the calculations considerably.