Classifying Spaces for Diffeological Groups

Fix an irrational number A, and consider the action of the group of pairs of integers on the real line defined as follows: the pair (m,n)(m,n) sends a point x to x+m+nA. 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.<br><br>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 A and B, the resulting irrational tori are diffeomorphic if and only if there is a fractional linear transformation with integer coefficients relating A and B, 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, the main topic of this talk.<br><br>We will perform Milnor's construction in the realm of diffeology to obtain a diffeological classifying space for a diffeological group G, such as the irrational torus. After mentioning a few hoped-for properties, we then construct a connection 1-form on the G-bundle EG→BG, which will naturally pull back to a connection 1-form on sufficiently nice principal G-bundles. We then look at what this can tell us about irrational torus bundles. (Joint work with Jean-Pierre Magnot)