K theory of the wedge of circles I am interested in finding the $K_0, K_1$ groups of $C(S^1 \vee S^1)$.
We know that $K_0 (C(S^1)) = K_1(C(S^1)) = \mathbb Z$, but is this directly helpful?
 A: If $X$ is a compact space written as $X=A\cup B$, where $A$ and $B$ are closed subsets, with $I=A\cap B$, then the Mayer-Vietoris sequence is an exact sequence of topological $K$-groups
$\require{AMScd}$
\begin{CD}
  K_0(C(X)) @>({\rho_A}_*,{\rho_B}_*)>> K_0(C(A))\oplus K_0(C(B)) @>{\pi_A}_*-{\pi_B}_*>> K_0(C(I))\\
  @A \partial  AA  @.
 @VV\text{ind}V\\
K_1(C(I))
@<<{\pi_A}_*-{\pi_B}_*<
K_1(C(A))\oplus K_1(C(B))
@<<({\rho_A}_*,{\rho_B}_*)<
K_1(C(X))
  \end{CD}
where

*

*$\rho_A:C(X)\to C(A)$,


*$\rho_B:C(X)\to C(B)$,


*$\pi_A:C(A)\to C(I)$,


*$\pi_B:C(B)\to C(I)$,
are the restriction maps.  Regarding the case in point, where $A=B=S^1$, and $I=\{*\}$, we have
\begin{CD}
 K_0(C(X))@>({\rho_A}_*,{\rho_B}_*)>> \mathbb Z\oplus \mathbb Z @>{\pi_A}_*-{\pi_B}_*>> \mathbb Z\\
  @A \partial  AA  @.
 @VV\text{ind}V\\
0
@<<{\pi_A}_*-{\pi_B}_*<
\mathbb Z\oplus \mathbb Z
@<<({\rho_A}_*,{\rho_B}_*)<
K_1(C(X))
  \end{CD}
so we must have that $K_0(C(X))=\mathbb Z$, and $K_1(C(X))=\mathbb Z\oplus \mathbb Z$.
