The diffeomorphism groups of the real line are pairwise bihomeomorphic

TL;DR

This paper proves that the diffeomorphism group of the real line with C^r regularity is topologically equivalent to the homeomorphism group, revealing a surprising similarity in their topological structures.

Contribution

It establishes a bihomeomorphism between the diffeomorphism group D^r(R) and the homeomorphism group H(R), showing their topological equivalence under certain topologies.

Findings

Diffeomorphism groups are bihomeomorphic to homeomorphism groups.

D^r(R) with Whitney topology is homeomorphic to a countable box-power of a Hilbert space.

Topological structures of diffeomorphism and homeomorphism groups are fundamentally similar.

Abstract

We prove that the group D^r(R) of C^r diffeomorphisms of the real line, endowed with the compact-open and Whitney C^r topologies, is bihomeomorphic to the group H(R) of homeomorphisms of the real line endowed with the compact-open and Whitney topologies. This implies that the diffeomorphism group D^r(R) endowed with the Whitney C^r topology is homeomorphic to the countable box-power of the separable Hilbert space.