**Some Background:** In Kontsevich and Soibelman's theory of motivic DT-invariants, one is interested in something like the ``number'' of objects in a 3-Calabi-Yau category $\mathcal{C}$ having a fixed class $\gamma$ in $K_0(\mathcal{C})$. One of the motivating examples is in counting the number of certain Lagrangian submanifolds (with some extra data) in a Calabi-Yau 3-fold $X$ with a fixed class in $H_3(X)$. In other words, we want to count objects in the Fukaya category $\mathcal{D}^bLag(X)$ of $X$ subject to some constraint.

Question 1:Is it known (that under certain circumstances) that $K_0(\mathcal{D}^bLag(X)$) is isomorphic to $H_3(X)$?

I guess while we are at it, what about the higher $K$-groups, or the $K$-theory spectrum. My suspicion is not much is known, since I believe that very little is known about the higher $K$-theory of triangulated categories in general. Any comments or references would be much appreciated.