The K-Theory of Heegaard-Type Quantum 3-Spheres

Abstract

We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C * -algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C * -algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts.

Piecewise principal comodule algebras

“…To end with, we recall from [DHH], [HMS06a], [BHMS05] and [HMS06b] certain constructions of algebras and show that they are piecewise trivial comodule algebras. This way we indicate possible areas of applications of Section 3.…”

“…Based on the idea of a Heegaard splitting of S 3 into two solid tori, a noncommutative deformation of S 3 was proposed in [CM02], [HMS06a], [BHMS05]. On the level of C*-algebras, it can be presented as a fibre product C.S The identification is given by T Ì Â Z 1 2 % % P P P P P P P P P P…”

On piecewise trivial hopf—Galois extensions

“…Let H be a Hopf algebra. An H-Galois extension is an algebra extension B ⊂ P , where P is an H-comodule algebra with coaction ρ : P → P ⊗ H, B = P H-inv := {a ∈ P | ρ(a) = a ⊗ 1} is the subalgebra of H-invariants, and the canonical map (1) can : P ⊗ B P → P ⊗ H, a ⊗ a → (a ⊗ 1)ρ(a )…”

“…But one can also allow for more noncommutativity and work with crossed products B H whenever B can be equipped with the structure of an H-module algebra. This is needed for example to cover examples like the Heegaard-type quantum 3-spheres [1,9].…”

“…Principal extensions obtained by gluing trivial extensions such as the Heegaard-type quantum 3-spheres [1,8,9] provide nice examples that will be studied in detail in [11]. On the other hand, B might not admit a covering at all which happens if it is an integral domain, or a simple algebra or a primitive C * -algebra.…”