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

TL;DR

This paper classifies differentiable structures on a non-Hausdorff line with two origins, revealing uncountably many non-diffeomorphic structures for each differentiability class.

Contribution

It provides a complete classification of differentiable structures on the line with two origins using double coset classes, a novel approach in this context.

Findings

Uncountably many non-diffeomorphic ^{k}-structures for each k.

A bijection between ^{k}-structures and double coset classes.

Distinct differentiable structures compared to the real line.

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 $\mathbb{R}\setminus0$ 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 $D \setminus H / D = \{ D h D \mid h \in H\}$. 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 $D \setminus H^{\pm} / D = \{ D h D \cup D h^{-1} D \mid h \in H\}$. 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 $k=1,2,\ldots,\infty$.