Fix an irrational number , and consider the action of the group of pairs of integers on the real line defined as follows: the pair sends a point to . The orbits of this action are dense, and so the quotient topology on the orbit space is trivial. Any reasonable notion of smooth function on the orbit space is constant. However, the orbit space is a group: the orbits of the action are cosets of a normal subgroup. Can we give the space any type of useful "smooth" group structure?
The answer is "yes": its natural diffeological group structure. It turns out this is not just some pathological example. Known in the literature as the irrational torus, as well as the infra-circle, this diffeological group is diffeomorphic to the quotient of the torus by the irrational Kronecker flow, it has a Lie algebra equal to the real line, and given two irrational numbers and , the resulting irrational tori are diffeomorphic if and only if there is a fractional linear transformation with integer coefficients relating and , and so it is of interest in many fields of mathematics. Moreover, it shows up in geometric quantisation and the integration of certain Lie algebroids as the structure group of certain principal bundles.
Because of this, I am interested in bundles with the irrational torus as the structure group. One nice way to study principal bundles is the classifying space of the structure group. In [MW], we construct the classifying space of a diffeological group using Milnor's construction, and prove many hoped-for properties. We also build an almost universal connection of sorts on , which pullsback to a connection on a diffeological fibre bundle, in the sense of Iglesias-Zemmour [IZ].