Linear Microbundles

TL;DR

This paper extends calculus to topological manifolds using generalized functions, enabling the construction of non-singular tangent vector fields on spheres of any dimension, and discusses broader applications of this theory.

Contribution

It introduces a generalized calculus framework for topological manifolds, broadening the scope of differential calculus beyond smooth structures.

Findings

Existence of non-singular generalized tangent vector fields on spheres of any dimension.

Coherence with the non-existence of smooth vector fields on even-dimensional spheres.

Potential applications in topology and geometric analysis.

Abstract

We extend calculus from smooth manifolds to topological manifolds making use of a theory of generalized functions developed for this aim. Actually such extension fits into a boarder context: the universal construction of a site containing all continuous maps between topological spaces and whose arrows are smooth in the generalized sense. As an application we prove the existence of non singular generalized tangent vector fields on spheres of any dimension, showing how this result is coherent with the non existence of smooth vector fields on spheres of even dimension. Many other applications are outlined or suggested, some of which are under development by the author.