Exotic Differential Structures in Dimension 2

TL;DR

This paper demonstrates that the long plane admits a vast number of exotic differential structures, expanding the understanding of differential topology in higher-dimensional non-metrisable manifolds.

Contribution

It establishes the existence of many exotic differential structures on the long plane, analogous to known results for the long line, revealing new complexity in non-metrisable manifolds.

Findings

The long plane supports $2^{eth_1}$ many non-diffeomorphic structures.

Exotic structures are not just products of structures on factors.

The result parallels known phenomena in the long line.

Abstract

It is known that the long line supports $2^{\aleph_1}$ many non-diffeomorphic differential structures. We show that the long plane supports a similar number of exotic differential structures, ie structures which are not merely diffeomorphic to the product of two structures on the factor spaces.