We introduce $\varepsilon$-approximate versions of the notion of Euclidean vector bundle for $\varepsilon \geq 0$, which recover the classical notion of Euclidean vector bundle when $\varepsilon = 0$. In particular, we study \v{C}ech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon$-approximate vector bundles can be used to represent classical vector bundles when $\varepsilon > 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data, and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

中文翻译：

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近似和离散欧几里得向量束

我们引入$ \ varepsilon \ geq 0 $的$ \ varepsilon $-欧几里得向量束概念的近似版本，当$ \ varepsilon = 0 $时，它们恢复了欧几里得向量束的经典概念。特别地，我们研究满足正交近似条件的正交组中系数为\ v {C} ech的共链。我们显示，当$ \ varepsilon> 0 $足够小时，可以使用$ \ varepsilon $-近似向量束表示经典向量束。我们还介绍了近似向量束之间的距离，并使用它们来证明足够相似的近似向量束表示相同的经典向量束。这提供了一种使用有限数量的数据在有限简单复形上指定向量束的方法，并且在应用的设置中处理矢量束时还允许一定的噪声容忍度。例如，我们证明了有限样本中向量束的重构定理。我们给出了直接从离散表示和近似表示有效计算维数维的低维特征类的算法，并通过计算示例说明了这些算法的用法。