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A secondary Chern-Euler class. (English) Zbl 0980.57012
From the introduction: Let \(\xi\) be a smooth oriented vector bundle, with \(n\)-dimensional fibre, over a smooth manifold \(M\). Denote by \(\widehat\xi\) the fibrewise one-point compactification of \(\xi\). The main purpose of this paper is to define geometrically a canonical element \(\Upsilon (\xi)\) in \(H^n(\widehat \xi,\mathbb{Q})\) \((H^n(\widehat \xi,\mathbb{Z}) \otimes{1\over 2}\), to be more precise). The element \(\Upsilon(\xi)\) 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 \(\xi\) is the tangent bundle \(TM\) of \(M\) (hence \(M\) is oriented). In this case we denote \(\widehat\xi\) by \(\Sigma M\) and simply write \(\Upsilon\) for \(\Upsilon (\widehat\xi)\).
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 \(\alpha:M\to \Sigma M\). One can extend \(V\) to a smooth tangent vector field \(\overline V\) 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 \(\overline V\), let \(\text{ind}_p (\overline V)\) be the index of \(\overline V\) at \(p\) defined as usual. We then have the following:
Theorem 0.1. For any generic extension \(\overline V\) of \(V\), if \(p_1,\dots,p_k\) are the zero points of \(\overline V\) then \[ \sum^k_{i=1} \text{ind}_{p_j} (\overline V)= \begin{cases} \chi(X) +\alpha^*(\Upsilon) \bigl([M]\bigr) \quad &\text{if }n\text{ is odd}\\ \alpha^* (\Upsilon) \bigl([M] \bigr)\quad &\text{if }n \text{ is even}\end{cases} \] where \(\chi(X)\) is the Euler characteristic of \(X\).
The second property of \(\Upsilon (\xi)\) is that it is closely related to the Thom class. Let \(\xi_\infty\) be the \(\infty\)-section of \(\widehat\xi\), and let \(\gamma(\xi)\in H^n(\widehat \xi, \xi_\infty)\), with integer coefficients, be the Thom class of \(\xi\). We shall show the following:
Theorem 0.2. The natural homomorphism \(j^*:H^n (\widehat \xi,\xi_\infty)\to H^n(\widehat\xi)\) is injective, and \(j^*(\gamma (\xi))=\Upsilon (\xi)+{1 \over 2}\sigma^* (e(\xi))\) where \(e(\xi)\) is the Euler class of \(\xi\), and \(\sigma:\widehat \xi\to M\) is the projection.

57R20 Characteristic classes and numbers in differential topology
53C99 Global differential geometry
57R25 Vector fields, frame fields in differential topology
58A99 General theory of differentiable manifolds
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