The generator of $K_0(C(\partial(]0,1[^2)))$ and $K_1(C(\partial(]0,1[^2)))$

Question

Let $C=[0,1]^2 \subseteq \mathbb{C}$ and $\partial C$ the boundary of $C$. I'm looking for the $K_0(C(\partial C))$ and $K_1(C(\partial C))$ and its generator.

Answer 1

From the split exact sequence $$0\rightarrow C_0(\mathbb R)\to C(S^1)\leftrightarrows\mathbb C\to 0,$$ we obtain split exact sequences in $K$-theory $$0\to K_i(C_0(\mathbb R))\to K_i(C(S^1))\leftrightarrows K_i(\mathbb C)\to 0,$$ so that $$K_i(C(S^1))=K_i(C_0(\mathbb R))\oplus K_i(\mathbb C)\cong\mathbb Z.$$ Now this tells us that $K_0(C(S^1))=K_0(\mathbb C)$ is generated by the class of identity function $z\mapsto 1$ in $C(S^1)$.

For the generator of $K_1(C(S^1))$, we first note that $K_1(C(S^1))=K_1(C_0(\mathbb R))\cong K_0(\mathbb C)$, where the last isomorphism is via the Bott map. Since the Bott map applied to $1\in\mathbb C$ gives us the identity function $z\mapsto z\in C(S^1)$, this tells us that $K_1(C(S^1))$ is generated by the class of this function.