$C^\infty$-manifolds with skeletal diffeology

TL;DR

This paper introduces the concepts of $d$-skeletal and $d$-coskeletal diffeologies, analyzing their properties on $C^$-manifolds, and reveals significant topological and homotopical differences from classical manifolds.

Contribution

It generalizes wire diffeology to $d$-skeletal diffeology, studying their topological properties and homotopy groups, and highlights pathological behaviors for certain dimensions.

Findings

$d$-skeletal diffeology preserves paracompactness and smoothness for finite-dimensional manifolds.

Immersions become isolated in the diffeological space of smooth maps for $d< ext{dim} ext{M}.

The $d$-dimensional smooth homotopy group of $M_d$ is uncountable.

Abstract

We formulate and study the notion of $d$-skeletal diffeology, which generalizes that of wire diffeology, introducing the dual notion of $d$-coskeletal diffeology. We first show that paracompact finite-dimensional $C^\infty$-manifolds $M_d$ with $d$-skeletal diffeology inherit good topologies and smooth paracompactness from $M$. We then study the pathology of $M_d$. Above all, we prove the following: For $d<{\rm dim}\ M$, every immersion $f:M\longrightarrow N$ is isolated in the diffeological space $\dcal(M_d, N_d)$ of smooth maps and the $d$-dimensional smooth homotopy group of $M_d$ is uncountable.