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On topological invariants of vector bundles. (English) Zbl 0765.57020

The purpose of this paper is to calculate the Euler number \(\chi(E)\) of an orientable vector bundle \(E\to W\) in terms of equations describing \(E\) and \(W\). More explicitly, let \(E\to W\) be an \((n-k)\)-dimensional vector bundle over \(W\), where \(W\) is an \((n-k)\)-dimensional, compact, \(C^ 1\)- manifold defined as \(F^{-1}(0)\) for \(F:R^ n\to R^ k\) a \(C^ 1\)-map with rank \(| DF(x)|=k\) at all \(x\in W\), and where \(E=\{(x,y)\in W\times R^ m| y\perp G_ i(x)\), where \(G_ i:R^ n\to R^ m\) is a family of \(C^ 1\)-functions, \(i=1,\dots,s\), \(m=s+n-k\), with \(G_ 1(x),\dots,G_ s(x)\) linearly independent for all \(x\in W\}\). If \(H:R^ n\times R^ s\to R^ m\times R^ k\) is given by \(H(x,\lambda)=\left(\sum^ s_{i=1}\lambda_ iG_ i,\;F(x)\right)\), and \(S_ r=\{(x,\lambda)\in R^ n\times R^ s|\| x\|^ 2+\|\lambda\|^ 2=r^ 2\}\), where \(r>0\) is such that \(W\subset\{x\in R^ n|\| x\|<r\}\) then the topological degree, \(\deg(H| S_ r)\), of \(H| S_ r:S_ r\to R^ m\times R^ k- \{0\}\) is well defined. It is proved that \(\chi(E)=(- 1)^{n(s+k)+k}\deg(H| S_ r)\) and that \(\chi(W)=(-1)^ k\deg(H| S_ r)\) when \(H\) is calculated with \(s=k\) and \(G_ i=\text{grad} F_ i\), \(i=1,\dots,k\).

MSC:

57R20 Characteristic classes and numbers in differential topology
57R22 Topology of vector bundles and fiber bundles
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