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 C^k-diffeomorphism. Let also D be the subgroup of H consisting of C^k-diffeomorphisms of R also fixing 0. It is shown that there is a natural bijection between C^k-structures on L (up to a C^k-diffeomorphism fixing both origins) and double D-coset classes D H / D = \ D h D h H\. Moreover, the set of all C^k-structures on L (up to a C^k-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 C^k-structures for each k=1, 2, ,.