Topological (\prod^\omega \ell_2, \sum^\omega \ell_2)-factors of diffeomorphism groups of non-compact manifolds

TL;DR

This paper demonstrates that the pair of diffeomorphism groups of a non-compact manifold admits a specific topological factorization, leading to a classification of their topological structure, especially in dimension two.

Contribution

It establishes a topological ( extprod^\omega \ell_2, extsum^\omega \\ell_2)-factorization for diffeomorphism groups of non-compact manifolds and characterizes their topological types in dimension two.

Findings

The pair (D(M), D^c(M)) admits a topological ( extprod^\omega \\ell_2, extsum^\omega \\ell_2)-factor.

In dimension two, (D(M)_0, D^c(M)_0) forms a ( extprod^\omega \\ell_2, extsum^\omega \\ell_2)-manifold.

Results extend to groups of homeomorphisms of non-compact topological 2-manifolds.

Abstract

Suppose M is a non-compact connected smooth n-manifold. Let D(M) denote the group of diffeomorphisms of M endowed with the compact-open C^\infty-topology and D^c(M) denote the subgroup consisting of diffeomorphisms of M with compact support. Let D(M)_0 and D^c(M)_0 be the connected components of id_M in D(M) and D^c(M) respectively. In this paper we show that the pair (D(M), D^c(M)) admits a topological (\prod^\omega \ell_2, \sum^\omega \ell_2)-factor. In the case n = 2, this enables us to apply the characterization of (\prod^\omega \ell_2, \sum^\omega \ell_2)-manifolds and show that the pair (D(M)_0, D^c(M)_0) is a (\prod^\omega \ell_2, \sum^\omega \ell_2)-manifold and determine its topological type. We also obtain a similar result for groups of homeomorphisms of non-compact topological 2-manifolds.