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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.

##### MSC:
 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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