An Interpretation of G. Whitehead's Generalization of H. Hopf's Invariant

Abstract

In the present paper', the generalized H. Hopf's invariant H: 7rdwn++(Sn+l) w7rd +n+i(Sqn+ ), due to G. Whitehead [10], is given a new definition which has some similarity with the original H. Hopf's definition [5]. The invariant H(f ) appears as depending in particular on the position in Sd+n+l of the inverse images M, Md byf: Sdln+l -_ Sn+1 of two regular values q, q' in Sn+1. Arnold Shapiro has defined the linking coefficient of two spheres Sp, Sq imbedded (without common point) in Em+i for p + q > m.2 In ? 5, the notion of linking coefficient is extended to (p + q m)-connected 7rmanifolds Mp, Mq in Em+i. It is an element of the stable homotopy group 7rW+N(SN), where r p + q m (7r-manifold = manifold which can be imbedded in some euclidean space with a trivial normal bundle). In the definition of H given in ? 3, M, and M' are 7r-manifolds but need not be (d n)-connected. As a consequence, H will in general also depend on the fields of normal vectors over M and M'. Therefore, it cannot be considered strictly as a linking coefficient which should be uniquely determined by the position in space of the two manifolds. It is an open question whether the method can be used to define the linking coefficient of (non-necessarily (p + q m)-connected) w-manifolds Mp, Mq in Em+i by going over to the quotient of 7Wp+qm,+N(SN) by some suitable subgroup. As an application of the new definition of H, it is proved that any regular imbedding (without self-intersection) of the d-sphere into euclidean (d + n)-space induces over Sd the trivial normal bundle provided that 2n > d + 1. A partial result in this direction was announced in [6].