Manifolds and discrete structures (Q791935)

The starting point of this paper is the connection between smooth structures on a manifold M, the algebra of smooth functions on M and the behaviour of smooth functions on a dense subset of M. It is shown that if \((a_ n)\) and \((b_ n)\) are dense sequences in M with \(a_ m\neq a_ n\) and \(b_ m\neq b_ n\) for \(m\neq n\) then there is a diffeomorphism \(\phi\) : \(M\to M\) with \(\phi(\{a_ n| n=1,...,\infty \})=\{b_ n| n=1,...,\infty \}\). A discrete structure on M is an equivalence class of dense sequences, where two sequences \((a_ n)\) and \((b_ n)\) as above are equivalent if there is a diffeomorphism \(\phi\) of M and a positive integer N such that for each \(n\geq N\), \(\phi(a_ n)=b_ n\). There is a natural relationship between the set of discrete structures on M and a certain coset space of the symmetric group on the positive integers. A condition under which two sequences give rise to the same discrete structure is given.