On topological invariants of vector bundles

Abstract

Let E →W be an oriented vector bundle, and let X(E) denote the Euler number of E. The paper shows how to calculate X(E) in terms of equations which describe E and W . Introduction. Let F = (F1, . . . , Fk) : R → R, n − k > 0, be a C-map such that W = F−1(0) is compact and rank[DF (x)] ≡ k at every x ∈ W . From the implicit function theorem W is an (n − k)-dimensional C-manifold. Let G1, . . . , Gs : R→R, where m = s+n−k, be a family of C-vector functions, and assume that the vectors G1(x), . . . , Gs(x) are linearly independent for every x ∈W . Define E = {(x, y) ∈W × R | y ⊥ Gi(x), i = 1, . . . , s} = { (x, y) ∈W × R ∣∣∣ ∑ yjGji (x) = 0, i = 1, . . . , s} . Clearly E is an (n− k)-dimensional vector bundle over W . In particular, if s = k and Gi = gradFi then E becomes TW . Later we shall describe how to orient W and E. Let X(E) be the Euler number of the bundle E (see [1], Chapter 5.2). The problem is how to calculate X(E) in terms of F and G1, . . . , Gs. Let SR = {(x, λ) ∈ R×R | ‖x‖ +‖λ‖ = R}, and let H : R×R → R × R be the map given by