Differentiable structures on a union of two open sets

TL;DR

This paper classifies differentiable structures on a non-Hausdorff one-dimensional manifold called the letter "Y", revealing uncountably many non-diffeomorphic structures, and extends the classification approach to a categorical framework.

Contribution

It provides a classification of differentiable structures on the non-Hausdorff manifold "Y" and generalizes the classification method to categorical contexts.

Findings

Both manifolds admit uncountably many non-diffeomorphic ^k-structures.

Classification proofs are similar and extendable to categorical arrows.

Contrast with the line with two origins, which also admits uncountably many structures.

Abstract

In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold $\mathbb{L}$ called the line with two origins which is obtained by gluing two copies of the real line $\mathbb{R}$ via the identity homeomorphism of $\mathbb{R}\setminus 0$. Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold $\mathbb{Y}$ (called letter "$Y$") obtained by gluing two copies of $\mathbb{R}$ via the identity map of positive reals. It turns out that, in contrast to the real line, for every $r=1,\ldots,\infty$, both manifolds $\mathbb{L}$ and $\mathbb{Y}$ admit uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures. We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories.