Using a certain well-posed ODE problem introduced by Shilnikov in the sixties, Minervini proved the currential “fundamental Morse equation” of Harvey–Lawson but without the restrictive tameness condition for Morse gradient flows. Here, we construct local resolutions for the flow of a section of a fiber bundle endowed with a vertical vector field which is of Morse gradient type in every fiber in order to remove the tameness hypothesis from the currential homotopy formula proved by the first author. We apply this to produce currential deformations of odd degree closed forms naturally associated to any hermitian vector bundle endowed with a unitary endomorphism and metric compatible connection. A transgression formula involving smooth forms on a classifying space for odd K-theory is also given.

Minervini 利用 Shilnikov 在六十年代引入的一个适定 ODE 问题，证明了当前 Harvey-Lawson 的“基本 Morse 方程”，但没有 Morse 梯度流的限制性驯服条件。在这里，我们为每根纤维中具有莫尔斯梯度类型的垂直矢量场的纤维束的一段流动构造局部分辨率，以消除第一作者证明的当前同伦公式中的驯服假设。我们应用它来产生奇数度封闭形式的当前变形，这些形式自然地与任何具有酉自同态和度量兼容连接的厄尔米特向量束相关联。奇数K分类空间上涉及平滑形式的越界公式-还给出了理论。