UC Berkeley

Abstract: We investigate the ``$\theta$-deformed spheres" $C(S^{3}_{\theta})$ and $C(S^{4}_{\theta})$ for the case $\theta$ an irrational number. We show that all finitely-generated projective modules over $C(S^{3}_{\theta})$ are free, and that $C(S^{4}_{\theta})$ has the cancellation property. We classify and construct all finitely-generated projective modules over $C(S^{4}_{\theta})$ up to isomorphism. An interesting feature is that there are nontrivial ``rank-1" modules over $C(S^{4}_{\theta})$. Every finitely-generated projective module over $C(S^{4}_{\theta})$ is a sum of rank-1 modules. This is because the group of path-components of the invertible elements of $C(S^{3}_{\theta})$ is $\mathbb{Z}$ and maps isomorphically onto $K_{1}(C(S^{3}_{\theta}))$ under the natural map.