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

Abstract

We classify differentiable structures on a line $\mathbb{L}$ with two origins being a non-Hausdorff but $T_1$ one-dimensional manifold obtained by ``doubling'' $0$. For $k\in \mathbb{N}\cup \{\infty \}$ let $H$ be the group of homeomorphisms $h$ of $\mathbb{R}$ such that $h(0)=0$ and the restriction of $h$ to $\mathbb{R}\setminus 0$ is a ${\mathcal{C}}^k$-diffeomorphism. Let also $D$ be the subgroup of $H$ consisting of ${\mathcal{C}}^k$-diffeomorphisms of $\mathbb{R}$ also fixing $0$. It is shown that there is a natural bijection between ${\mathcal{C}}^k$-structures on $\mathbb{L}$ (up to a ${\mathcal{C}}^k$-diffeomorphism fixing both origins) and double $D$-coset classes $D\setminus H/D=\{DhD\mid h\in H\}$. Moreover, the set of all ${\mathcal{C}}^k$-structures on $\mathbb{L}$ (up to a ${\mathcal{C}}^k$-diffeomorphism which may also exchange origins) are in one-to-one correspondence with the set of double $(D,\pm )$-coset classes $D\setminus H^{\pm }/D=\{DhD\cup D{h}^{-1}D\mid h\in H\}$. In particular, in contrast with the real line, the line with two origins $\mathbb{L}$ admits uncountably many pair-wise non-diffeomorphic ${\mathcal{C}}^k$-structures for each $k=1,2,\dots ,\infty$.

29 pages, 3 figures