On the geometry of profinite diffeological spaces

TL;DR

This paper explores the structure of profinite diffeological spaces, analyzing their geometric and topological properties, including tangent spaces, differential forms, and cohomology, to unify various geometric constructions.

Contribution

It introduces a comprehensive framework for understanding profinite diffeological spaces, connecting differential geometry, topology, and algebraic structures in this context.

Findings

Analysis of tangent and cotangent spaces

Development of differential forms and Riemannian metrics

Relation between de Rham and singular cohomology

Abstract

We consider the class of profinite diffeological spaces, that is, diffeological spaces which diffeologies are deduced by pull-back of diffeologies on finite-dimensional manifolds through a system of projection mappings. This class includes inductive limits of finite-dimensional manifolds, as well as solution spaces of differential relations and spaces that agree with cylindrical approximations. We analyze tangent and cotangent spaces, differential forms, Riemannian metrics, connections, differential operators, Laplacians, symplectic forms, momentum maps and the relation between de Rham and singular cohomology on them, unifying constructions partially developed in various contexts.