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

The Whitney group $K_1(A)$ is isomorphic to $A^\times \times \operatorname{SK}_1(A)$forsomesubgroup$\operatorname{SK}_1(A)$,where$A$isacommutativeringand$A^\times$denotesthesetofunitsin$A$. Consider an o-minimal expansion of a real closed field $\mathcal R=(R,0,1,+,\cdot,\ldots)$\mathrm{.}Let$M$beanaffinedefinable$C^r$manifold,where$r$ is a nonnegative integer. We demonstrate its homotopy theorem and that the group ${\mathrm{SK}}_1(C_{\text{df}}^r)$ is isomorphic to ${\mathrm{SK}}_1(C_{\text{df}}^0(M))$, where $C_{\text{df}}^r(M)$ denotes the ring of definable $C^r$ functions on $M$.