Whitehead group of the ring of smooth functions definable in an o-minimal structure

Abstract

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 $M$.