Topological (\prod^ω\ell_2, \sum^ω\ell_2)-factors of diffeomorphism groups of non-compact 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 ∞ -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 idM in D(M ) and D c (M ) respectively. In this paper we show that the pair (D(M ), D c (M )) admits a topological ω ℓ2, ω ℓ2 -factor. In the case n = 2, this enables us to apply the characterization of ω ℓ2, ω ℓ2 -manifolds and show that the pair (D(M )0, D c (M )0) is a ω ℓ2, ω ℓ2 -manifold and determine its topological type. We also obtain a similar result for groups of homeomorphisms of non-compact topological 2-manifolds.