Department of Mathematics


Differential Structures on Large Manifolds

A differential structure on a manifold provides a way of transferring techniques of differential calculus to the more general context of a manifold.

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A huge amount of effort has been expended in studying small, especially compact, manifolds and their differential structures. We are researching differential structures on manifolds that are so large that they do not support a metric (distance function).

A manifold is a topological space which, to a sufficiently short-sighted observer, looks like Euclidean space of some dimension: in dimension 1 it looks locally like a line, and examples include a line itself and a circle, in dimension 2 it looks locally like a very thin sheet of paper and examples include the surfaces of a sphere, a torus and the Klein bottle.

A differential structure on a manifold M is a collection C consisting of pairs (U,φ), where U is an open subset of M and φ is a homeomorphism from U to the appropriate euclidean space so that if (U,φ) and (V,ψ) are two members of C then ψφ-1 is smooth where defined.

It has been known for two decades that the long line, a line which is so long that no sensible notion of distance can be assigned to it, supports uncountable distinct differential structures, thus the same applies to the square of the long line. We are exploring whether there are differential structures on the square of the long line which are distinct from product structures.

Researchers at The University of Auckland