Immersions of real projective spaces into complex projective spaces
Tsutomu Yasui · Hiroshima Mathematical Journal · 1987
Given two differentiable manifolds M, N and a continuous map /: M-»N, we denote by /[M, N~\j the set of regular homotopy classes of immersions of M into ΛΓ, which are homotopic to/, and by /[M, N] the set of all regular homotopy classes of immersions of M into N.As for the set /[M, ΛΓ], some results have so far been obtained when N is not Euclidean space.For example, for the existence of immersions of P"(Q)and of L"(p) into L m (p), see [1], [4] and [15], [8], and [5], respectively, and for the classification of immersions of P"(R) into P m (R) 9 see [7]-[9], where P k (F) is the F-projective space of F-dimension k and L k (p) is the lens space mod p.But the above results are smaller in number than those when N is Euclidean space.In this article we shall study the set /[P M (K), P m (C)] and I[P"(R), P m (C)~] f for any map/: P n (R)->P m (C) (n^2m-\).Here we note the fact that [P"(#), P m (C)]=Z 2 if n^2m (see (2.5) below).Let i: P"(R)-+P"(C)c:P m (C) (n^ni) be the natural embedding defined by regarding real numbers as complex numbers and let c: P n (R)-+P m (C) be a constant map.Then we shall prove the following theorems :