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

Abstract

The Whitney group $K_1(A)$ is isomorphic to $A^{\times }\times {\mathrm{SK}}_1(A)$ for some subgroup ${\mathrm{SK}}_1(A)$, 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 $\mathcal{R}=(R,0,1,+,\cdot ,\dots )$. Let $M$ be an affine definable $C^r$ 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 $M$.