Homotopy types of Diffeomorphism groups of noncompact 2-manifolds

TL;DR

This paper classifies the homotopy types of the identity component of diffeomorphism groups of noncompact 2-manifolds, revealing they are often contractible or homotopy equivalent to a circle, depending on the manifold.

Contribution

It provides a complete topological classification of the identity components of diffeomorphism groups for all noncompact 2-manifolds, including volume-preserving cases.

Findings

D(M)_0 is a topological ell_2-manifold.

D(M)_0 has the homotopy type of a circle for certain manifolds.

D(M)_0 is contractible for other noncompact 2-manifolds.

Abstract

Suppose M is a noncompact connected smooth 2-manifold without boundary and let D(M)_0 denote the identity component of the diffeomorphism group of M with the compact-open C^infty-topology. In this paper we investigate the topological type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has the homotopy type of the circle if M is the plane, the open annulus or the open M"obius band, and it is contractible in all other cases. When M admits a volume form w, we also discuss the topological type of the group of w-preserving diffeomorphisms of M. To obtain these results we study some fundamental properties of transformation groups on noncompact spaces endowed with weak topology.