Negative-degree complex K-groups

For a compact Hausdorff space $X$, let $X_{+}=X\bigsqcup \{\ast \}$ with the added point as basepoint. For $n\ge 0$, define

$K^{-n}(X)={\tilde{K}}^0({\mathrm{\Sigma }}^n{X}_{+})\mathrm{.}$

For a well-pointed based compact Hausdorff CGWH space $X$, define

${\tilde{K}}^{-n}(X)={\tilde{K}}^0({\mathrm{\Sigma }}^{n}X),$

and for a compact Hausdorff CGWH based pair $(X,A)$ whose quotient $X\mathrm{/}A$ is well-pointed define

$K^{-n}(X,A)={\tilde{K}}^0({\mathrm{\Sigma }}^n(X\mathrm{/}A))\mathrm{.}$

Suspensions and quotients use Reduced cone suspension and cofiber sequence and the based CGWH conventions of Compactly generated based spaces and well-pointed objects. At $n=0$, restriction to the added point splits $K^0(X_{+})\cong K^0(X)\oplus K^0(\ast )$; its kernel is canonically $K^0(X)$. Thus $K^0(X)$ in this grading agrees with the original unreduced group. This also covers $X=\mathrm{\varnothing }$, since $X_{+}=\ast$ and both groups are zero.