K-Thy mapping cone in pullback diagram

2012-05-27T01:00:29Z

Let $X, Y$ be connected, compact Hausdorff spaces and $\mathcal{H}_1, \mathcal{H}_2$ Hilbert spaces with $A \subset \mathcal{L}(\mathcal{H}_1), B \subset \mathcal{L}(\mathcal{H}_2)$ $\ast$-subalgebras. Set $\mathcal{K}_1 := \mathcal{K}(\mathcal{H}_1)$ for the compact operators and $\mathcal{K}_2 := \mathcal{K}(\mathcal{H}_2)$.

Assume that we have the canonical isomorphisms $A / \mathcal{K}_1 \cong C(X), B / \mathcal{K}_2 \cong C(Y)$ (via the quotient maps).

Let $\Sigma$ be a $C^{\ast}$-algebra pullback: \[ \Sigma := \{(F, G) \in C(X, B) \oplus C(Y, A) : q_Y(F) = q_X(G)\} \]

with the maps $$ q_Y \colon C(X, B) \to C(X \times Y), q_X \colon C(Y, A) \to C(X \times Y) $$

which are the pointwise quotient maps.

We can write down the Mayer-Vietoris theorem in K-theory and then we have the middle terms $K_i(C(X, A)) \oplus K_i(C(Y, B))$ in this sequence.

I would prefer $K^i(C(X)) \oplus K^i(C(Y)) \cong K^i(X) \oplus K^i(Y)$, i.e. a six-term exact sequence in K-theory no longer depending on $A, B$ and nontrivial examples of such situations.

I have the following observations (question at the end):

The mapping cone of $q_Y$ is defined as follows $$ C q_Y := \{(F, f) \in C(X, B) \oplus C C(X \times Y) : f(0) = q_Y(F)\} \ $$

for $q_X$ $$ C q_X := \{(G, g) \in C(Y, A) \oplus C C(X \times Y) : g(0) = q_X(G)\} $$

Note that with the short exact sequence $$ 0 \to C(X, \mathcal{K}_2) \to C q_Y \to C C(X \times Y) \to 0 $$

we have the isomorphism $K_i(C q_Y) \cong K_i(\ker q_Y) = K^i(X)$ and similarly for $q_X$.

Now define the (special) cone $$ C_{\Sigma} := \{(F, G, f) \in \Sigma \oplus C C(X \times Y) : f(0) = q_Y(F) = q_X(G)\} $$

We have an injective homomorphism $\kappa \colon C_{\Sigma} \to C q_X \oplus C q_Y, (F, G, f) \mapsto (F, f) \oplus (G, f)$.

Consider the short exact sequence $$ 0 \to SC(X \times Y) \to C_{\Sigma} \to \Sigma \to 0 $$

Where $j \colon SC(X \times Y) \to C_{\Sigma}, f \mapsto (0, 0, f)$ and $\varphi \colon C_{\Sigma} \to \Sigma, (F, G, f) \mapsto (F, G)$.

Now we can calculate the six-term exact sequence in K-theory. What I would like to have are isomorphisms in K-theory $K_i(C_{\Sigma}) \cong K_i(C q_X) \oplus K_i(C q_Y) \cong K^i(X) \oplus K^i(Y)$.

We would obtain a Mayer-Vietoris sequence which no longer depends on $A, B$.

But these cones are not homotopy equivalent. I would just like to know a concrete counterexample that such an isomorphism in K-theory does not hold (given $X, Y, A, B$ etc.).

K-Thy mapping cone in pullback diagram - MathOverflow

K-Thy mapping cone in pullback diagram