论文标题
可定义的$ c^r $矢量束和双线性的空间及其同型定理
Definable $C^r$ vector bundles and bilinear spaces in an o-minimal structure and their homotopy theorems
论文作者
论文摘要
考虑在真实场上的O最小结构。令$ m $为可定义的$ c^r $歧管，其中$ r $是一个非负整数。我们首先证明了$ m $上的可定义$ c^r $向量捆绑包的等效性与环$ c _ {\ text {df}}^r（m）$的有限生成的投射模块类别。在这里，符号$ c _ {\ text {df}}^r（m）$表示$ m $上的可定义$ c^r $ functions的戒指。我们还显示了$ m $上的可定义$ c^r $双线性空间的类别与环上的双线性空间类别$ c _ {\ text {df}}^r（m）$。本文的主要定理是可定义的$ c^r $ vector捆绑包和可定义的$ c^r $ bubinear spaces的同音定理。作为一个应用程序，我们表明grothendieck环$ k_0（c _ {\ text {df}}}^r（m））$，$ k_0（c _ {c _ {\ text {df}}}^0（m））
Consider an o-minimal structure on the real field. Let $M$ be a definable $C^r$ manifold, where $r$ is a nonnegative integer. We first demonstrate an equivalence of the category of definable $C^r$ vector bundles over $M$ with the category of finitely generated projective modules over the ring $C_{\text{df}}^r(M)$. Here, the notation $C_{\text{df}}^r(M)$ denotes the ring of definable $C^r$ functions on $M$. We also show an equivalence of the category of definable $C^r$ bilinear spaces over $M$ with the category of bilinear spaces over the ring $C_{\text{df}}^r(M)$. The main theorems of this paper are homotopy theorems for definable $C^r$ vector bundles and definable $C^r$ bilinear spaces over $M$. As an application, we show that the Grothendieck rings $K_0(C_{\text{df}}^r(M))$, $K_0(C_{\text{df}}^0(M))$ and the Witt ring $W(C_{\text{df}}^r(M))$ are all isomorphic.
