Exotic Differential Structures in Dimension 2

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.

💡 Research Summary

The paper investigates differential structures on the long plane, the Cartesian product of two copies of the long line, and demonstrates that it supports an enormous family of exotic structures, precisely $2^{\aleph_1}$ many, none of which are diffeomorphic to a simple product of differential structures on the factor lines. The authors begin by reviewing the classical result that the long line admits $2^{\aleph_1}$ mutually non‑diffeomorphic smooth structures, a consequence of the uncountable ordinal $\omega_1$ and the flexibility of transition maps on that space. They then describe the topology of the long plane $L\times L$, emphasizing that the naïve product of two long‑line structures yields only the “standard” smooth structure, which does not capture the full richness of possible smoothings.

The core contribution is a construction called a “convolutional twist.” For any pair of distinct smooth structures $S_\alpha$ and $S_\beta$ on the long line, the authors define a new atlas on $L\times L$ whose charts are products of basic charts from each factor, but whose transition maps are not simple products. Instead, each transition map includes an additional smooth cross term that depends on both coordinates and is supported on a carefully chosen $\omega_1$‑indexed interval. This cross term is $C^\infty$ and vanishes outside the interval, ensuring that the resulting transition maps remain smooth while introducing a genuine interaction between the two factors.

Two technical lemmas guarantee that these twisted transition maps compose smoothly and that the resulting atlas satisfies all the axioms of a smooth manifold. The first lemma exploits the limiting stability of the long‑line transition functions; the second establishes an equivalence relation on the set of twisted maps that preserves $C^\infty$ compatibility. By varying the pair $(\alpha,\beta)$ over all $2^{\aleph_1}$ possibilities, the construction yields $2^{\aleph_1}$ distinct smooth structures on the long plane. The authors then prove that none of these structures can be obtained by taking a product of two one‑dimensional smoothings, because any such product would force the cross term to be identically zero, contradicting the definition of the twisted atlas.

The paper concludes with several remarks. First, the result shows that the phenomenon of a vast number of exotic smoothings is not confined to one‑dimensional non‑metrizable manifolds; it persists in dimension two. Second, the convolutional twist method appears adaptable to higher‑dimensional long manifolds, suggesting a pathway to constructing exotic smoothings on $L^n$ for $n\ge3$. Finally, the authors discuss the implications for the classification problem of smooth structures on non‑paracompact manifolds, highlighting how ordinal‑indexed constructions can generate a rich hierarchy of smooth types far beyond the countable realm. This work thus expands the landscape of differential topology in the non‑metrizable setting and opens new directions for future research.