Infinite-dimensional degree theory, especially for Fredholm maps with positive index as developed with Tromba, is combined with Ramer’s unpublished thesis work on finite co-dimensional differential forms. As an illustrative example, the approach of Nicolaescu and Savale to the Gauss–Bonnet–Chern theorem for vector bundles is reworked in this framework. Other examples mentioned are Kokarev and Kuksin’s approach to periodic differential equations and to forced harmonic maps. A discussion about how such forms and their constructions and cohomology relate to constructions for diffusion measures on path and loop spaces is also included.

中文翻译：

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无限维度理论和Ramer的有限共维微分形式

无限维度理论，特别是对于与 Tromba 开发的具有正指数的 Fredholm 映射，与 Ramer 未发表的关于有限共维微分形式的论文工作相结合。作为一个说明性的例子，Nicolaescu 和 Savale 对向量丛的 Gauss-Bonnet-Chern 定理的方法在这个框架中被重新设计。提到的其他例子是 Kokarev 和 Kuksin 对周期微分方程和强制谐波映射的方法。还讨论了这些形式及其构造和上同调如何与路径和环路空间上的扩散测量构造相关。