On the Immersibility of Almost Parallelizable Manifolds

Abstract

1. M. W. Hirsch has shown that an almost parallelizable nmanifold is immersible in the Euclidean (n+k)-space R if [1. He has proved the result by making use of a result due to M. Kervaire that the Smale invariant of an immersion of n-sphere in R vanishes if n<=2k--2 [4. In this paper we shall prove the following Proposition 1. An almost parallelizable n-manifold is immersible in R if n 0 (mod 4). Proposition 2. If n=--0 (mod 4), an almost parallelizable n-manifold is in general not immersible in Rn/. In particular, Hirsch’s result is best possible for n=4. The authors wish to express their thanks to prof. K. Aoki and Prof. T. Kaneko for their many valuable suggestions and several discussions. 2. In the following discussions, all manifolds are considered as connected, orientabJe C manifolds. By immersion f" MRP we mean C map whose Jacobian matrix has rank n--dim M at each point of M. A homeomorphic immersion will be called imbedding. A manifold M will be called parallelizable if its tangent bundle is trivial, we say M is almost parallelizable if M--x .is parallelizable for some xeM’. M will be called =-manifold if M is imbedded in R+ (kn) with trivial normal bundle Since, a non-closed (i.e. non-compact or with boundary) almost parallelizable manifold is parallelizable, hence it is immersible in Rn+ (Theorem 6.3 of [2). Therefore we may consider only closed manifolds. Let o. denote the obstruction to the extension over M of the cross section over Mn--x; o is an element of zr_(SO(k)). Now let J: zrn_(SO(k))--->Zrn+_l(Sk) be the Hopf-Whitehead homomorphism, it is well known that J(o) --0. Moreover the result of J. F. Adams [6 implies that homomorphism J is injective for Zrn_x(SO(k))-Z. In the case of n0 (mod 4), _ (SO(k))--O or Z. according to whether n--3, 5, 6, 7 (mod 8) or n----l, 2 (mod 8) respectively. From this it follows that in the case of n3, 5, 6, 7 (mod 8), o-0. In the case n-l, 2 (mod 8), o is also zero, since it belongs to the kernel