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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 $A$. Consideran o-minimal expansion of a real closed field $\mathcalR=(R,0,1,+,\cdot,\ldots)$. Let $M$ be an affine definable $C^r$ manifold, where$r$ is a nonnegative integer. We demonstrate its homotopy theorem and that thegroup $\operatorname{SK}_1(C_{\text{df}}^r)$ is isomorphic to$\operatorname{SK}_1(C_{\text{df}}^0(M))$, where $C_{\text{df}}^r(M)$ denotesthe ring of definable $C^r$ functions on $M$.

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