Homotopy invariants in differential geometry. I

Abstract

Introduction. In this series of papers we will discuss homotopy invariants of differentiable maps f: M -? N in various situations within the framework of differential geometry. We shall be particularly interested in the case where f is an immersion. Our principle is simple. We use the fact that the pullbackf*W of c is a differentiable homotopy invariant where w is an arbitrary cohomology class always over the real number field, R. We note if F= M x I -1 N is a differentiable map with f=f, thenf*w belongs to the same cohomology class asfJ*w where I is the interval [0, 1] and ft is defined by ft(x)=F(x, t), x E M, t EI L There are many known examples beside various characteristic classes. To quote a few of them, letf be an immersion of the two-torus T2 into the complement of the diagonal set of the six dimensional euclidean space R6 considered as R3 x R3; f: T2 R3 x R3 A, where A is the diagonal set. The space R3 x R 3A is diffeomorphic with R4 x S2 where S2 is the two-sphere. Thus, if a) denotes the volume element of 52, a becomes a 2-dimensional cohomology class of R4 x S2 by pulling back with the projection onto S2 and gives us a homotopy invariantf*co. Consider T2 interpreted as the direct product S1 x S 1 of circles andf as the pair of two closed curves ci: SI --> R3 without intersection. Then f *w is nothing but the linking number of two closed -curves cl and c2 (up to a universal constant multiple) according to Gauss [8]. Another (but somewhat more extraneous) example, due to J. H. C. Whitehead [12], is the Hopf invariant, H(f), for a differentiable map f: S2n-1 _? Sn. Again, denoting the volume element of the n-sphere Sn by c, H(f) is given by the closed form 0 Af*w where 0 is any (n l)-form with dO =f*U. In this paperf will be an isometric immersion of a compact oriented Riemannian manifold M into a euclidean space Rn + v. Since the cohomology groups of N=Rn + v are then trivial, our principle does not apply tof directly, but we construct a manifold B from M and replace f by a certain map fB: M -? B. To do this, let us recall Hirsch's theorem [5] to the effect that the regular homotopy classes of the immersionsf: M -? Nofany manifold Minto another manifold N with dim N> dim M are in a one-to-one correspondence with the homotopy classes of the crosssections fB: M -->B of a certain bundle B over M, where a regular homotopy F= M x I -* N means one for which each ft is an immersion. Despite the triviality of the cohomology groups of N= Rn+ v in the case above, we can expect to obtain homotopy invariants fB*0 corresponding to a cohomology class 0 of B. In ?4, we