论文标题
Whitehead在O最低结构中可以定义的光滑函数环
Whitehead group of the ring of smooth functions definable in an o-minimal structure
论文作者
论文摘要
The Whitney group $K_1(A)$ is isomorphic to $A^\times \times \operatorname{SK}_1(A)$ for some subgroup $\operatorname{SK}_1(A)$, where $A$ is a commutative ring and $A^\times$ denotes the set of units in $A$.考虑一个真实闭合字段$ \ Mathcal r =（r，0,1，+，\ cdot，\ ldots）$的O-Minimal扩展。令$ m $为仿射定义的$ c^r $歧管，其中$ r $是一个非负整数。我们演示了其同质定理，并且组$ \ operatorName {sk} _1（c _ {c _ {\ text {df}}^r）$是与$ \ operatotorname {sk}} _1（c _ {c _ {c _ {\ text {df}}}^0（M）$， $ c _ {\ text {df}}^r（m）$表示$ m $上的可定义$ c^r $ functions的戒指。
The Whitney group $K_1(A)$ is isomorphic to $A^\times \times \operatorname{SK}_1(A)$ for some subgroup $\operatorname{SK}_1(A)$, where $A$ is a commutative ring and $A^\times$ denotes the set of units in $A$. Consider an o-minimal expansion of a real closed field $\mathcal R=(R,0,1,+,\cdot,\ldots)$. Let $M$ be an affine definable $C^r$ manifold, where $r$ is a nonnegative integer. We demonstrate its homotopy theorem and that the group $\operatorname{SK}_1(C_{\text{df}}^r)$ is isomorphic to $\operatorname{SK}_1(C_{\text{df}}^0(M))$, where $C_{\text{df}}^r(M)$ denotes the ring of definable $C^r$ functions on $M$.
