PI spaces with analytic dimension 1 and arbitrary topological dimension

TL;DR

This paper constructs metric measure spaces with arbitrary topological dimension n that are doubling, satisfy a Poincare inequality, and have a measurable tangent bundle of dimension 1, revealing new geometric measure theory phenomena.

Contribution

It introduces a method to build spaces with prescribed topological dimension and analytic tangent bundle dimension, expanding understanding of metric measure space structures.

Findings

Spaces with arbitrary topological dimension n and tangent bundle dimension 1 exist.

Constructed spaces satisfy doubling and Poincare inequality conditions.

The work demonstrates new possibilities for tangent bundle dimensions in metric measure spaces.

Abstract

For every n, we construct a metric measure space that is doubling, satisfies a Poincare inequality in the sense of Heinonen-Koskela, has topological dimension n, and has a measurable tangent bundle of dimension 1.