Let a finite group $G$ act on the circle $S^1$ via a group homomorphism $\varphi \colon G \to S^1$. Let $K = \ker \varphi$. I wish to calculate the equivariant K-theory group $K_G^*(S^1)$.
One method is to use the Mayer-Vietoris sequence. $S^1$ can be given a $G$-CW-complex structure with a single 0-cell $e^0 \times G/K$ and a single 1-cell $e^1 \times G/K$. Let $U$ be a small open neighbourhood around the 0-skeleton, and $V$ a small open neighbourhood around the complement of $U$. Visually, $V$ is a bit smaller then the interior of the 1-skeleton, so that it overlaps with $U$ just a little bit. Then $U$ and $V$ are homotopy equivalent to $G/K$ and $U \cap V$ is homotopy equivalent to two disjoint copies of $G/K$. With $S^1 = U \cup V$, and noting that $K^0_G(G/K) \cong R(K)$ and $K^1_G(G/K) = 0$, the Mayer-Vietoris sequence gives $$ 0 \to K^0_G(S^1) \to R(K)^2 \to R(K)^2 \to K^1_G(S^1) \to 0. $$ The center map is $(x,y) \mapsto (x-y, x-y)$. We conclude that $$ K^0_G(S^1) \cong R(K) \cong K^1_G(S^1). $$ All seems fine. As a sanity check, we apply Atiyah and Segal's fixed point decomposition theorem (from "On the equivariant Euler characteristic"). This tells us that $$ K_G(S^1) \otimes \mathbb C \cong \bigoplus_{[g]} ( K((S^1)^g) \otimes \mathbb C)^{C_g}, $$ where the direct sum is over the conjugacy classes of $G$, $(S^1)^g$ is the $g$-fixed points of $S^1$, and $C_g$ is the centraliser of $g$. Now, clearly $(S^1)^g$ is $S^1$ when $g \in K$ and empty otherwise. Thus, we get $$ K_G(S^1) \otimes \mathbb C \cong \bigoplus_{[g], g \in K} ( K(S^1) \otimes \mathbb C)^{C_g} \cong \bigoplus_{[g], g \in K} \mathbb C. $$ This is a complex vector space with dimension equal to the number of conjugacy classes of $G$ that are contained in $K$. This contradicts our Mayer-Vietoris calculation, since $R(K) \otimes \mathbb C$ has dimension equal to the number of conjugacy classes of $K$. Can anyone spot where I've made a mistake?