Combinatorial classification of quantum lens spaces

On the Classification of Fake Lens Spaces

In the first part of the paper we present a classification of fake lens spaces of dimension ≥ 5 whose fundamental group is the cyclic group of order any N ≥ 2. The classification is stated in terms of the simple structure sets in the sense of surgery theory. The results use and extend the results of Wall and others in the cases N = 2 and N odd and the results of the authors of the present paper...

Quantum Lens Spaces and Graph Algebras

We construct the C∗-algebra C(Lq(p;m1, . . . ,mn)) of continuous functions on the quantum lens space as the fixed point algebra for a suitable action of Zp on the algebra C(S2n−1 q ), corresponding to the quantum odd dimensional sphere. We show that C(Lq(p;m1, . . . ,mn)) is isomorphic to the graph algebra C∗ ( L (p;m1,...,mn) 2n−1 ) . This allows us to determine the ideal structure and, at lea...

Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

Similar to knots in S, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S and the similari...

Lens spaces in the Regge calculus approach to quantum cosmology.

Topological Invariants for Lens Spaces and Exceptional Quantum Groups

A pressing problem in the field of ‘quantum topology’ [1][2] is to understand the topological information embodied in the quantum invariants of 3-manifolds[3] [5] constructed in recent years, and to use the information to settle geometric questions. A direct way to tackle the problem is to compute these invariants for 3-manifolds of interest, then try to extract topological information from the...