Classification of differentiable structures on the non-Hausdorff line with two origins

We classify differentiable structures on a line L with two origins being a non-Hausdorff but T₁ one-dimensional manifold obtained by ``doubling'' $0$. For k∪\∞\ let H be the group of homeomorphisms h of R such that $h(0)=0$ and the restriction of h to R0 is a Cᵏ-diffeomorphism. Let also D be the subgroup of H consisting of Cᵏ-diffeomorphisms of R also fixing $0$. It is shown that there is a natural bijection between Cᵏ-structures on L (up to a Cᵏ-diffeomorphism fixing both origins) and double D-coset classes D H / D = \ D h D h ∈ H\. Moreover, the set of all Cᵏ-structures on L (up to a Cᵏ-diffeomorphism which may also exchange origins) are in one-to-one correspondence with the set of double (D,±)-coset classes D H± / D = \ D h D ∪ D h⁻¹ D h ∈ H\. In particular, in contrast with the real line, the line with two origins L admits uncountably many pair-wise non-diffeomorphic Cᵏ-structures for each k=1,2,…,∞.