A Secondary Chern-euler Class

Let ξ be a smooth oriented vector bundle, with n-dimensional fibre, over a smooth manifold M . Denote by ξ̂ the fibrewise one-point compactification of ξ. The main purpose of this paper is to define geometrically a canonical element Υ(ξ) in H(ξ̂,Q) (H(ξ̂,Z) ⊗ 12 , to be more precise). The element Υ(ξ) is a secondary characteristic class to the Euler class in the fashion of Chern-Simons. Two properties of this element are described as follows. The first one is in a very classical setting. Suppose ξ is the tangent bundle TM of M (hence M is oriented). In this case we denote ξ̂ by ΣM and simply write Υ for Υ(ξ̂). Suppose M is the boundary of a compact (n + 1)-dimensional smooth manifold X. Let V be a nowhere zero smooth vector field given on M which is tangent to X, but not necessarily tangent or transversal to M . The vector field V naturally defines a cross section α : M → ΣM . One can extend V to a smooth tangent vector field V on X with only isolated (hence only a finite number of) zeros. Since such extensions are generic we shall, for convenience, call any such extension a generic extension. At an isolated zero point p of V , let indp(V ) be the index of V at p defined as usual. We then have the following: