Kazuo Habiro

Quantum fundamental groups of 3-manifolds

Holonomy in the Hopf Bundle over the symmetric space \(\mathbb{C}P^n\)

video: comming soon.
The fundamental group of a basepointed topological space is one of the most fundamental concepts in algebraic topology. The fundamental group is important also in manifold topology, and in low dimensions it gives a very strong invariant.
In this talk, I plan to discuss the ``quantum fundamental group'' of a \(3\)-manifold, which is a refinement of the fundamental group, defined using isotopy classes of bottom tangles instead of homotopy classes of based loops. The set of bottom tangles in a \(3\)-manifold admits a very rich algebraic structure, and many constructions that work on fundamental groups work on quantum fundamental groups as well. For example, one can define the classifying space of the quantum fundamental group of a \(3\)-manifold, and the ``quantum representation variety'' associated to a co-ribbon Hopf algebra (such as the quantum group \(\mathrm{SL}_q(2)\)).