Diffeomorphism groups of non-compact manifolds endowed with the Whitney C^infty-topology

TL;DR

This paper studies the topological structure of diffeomorphism groups of non-compact manifolds with the Whitney C^infty-topology, revealing their homeomorphism types and how they relate to the manifold's homotopy type.

Contribution

It characterizes the homeomorphism types of identity components of diffeomorphism groups of non-compact manifolds, linking them to l_2-manifolds and their homotopy types.

Findings

DD_0(M) is homeomorphic to N imes IR^ infty, with N determined by the homotopy type.

For orientable, irreducible 3-manifolds, DD_0(M) is homeomorphic to l_2 imes IR^ infty.

The structure of DD_0(N - oundary N) relates to boundary-preserving diffeomorphisms.

Abstract

Suppose M is a non-compact connected n-manifold without boundary, DD(M) is the group of C^\infty-diffeomorphisms of M endowed with the Whitney C^\infty-topology and DD_0(M) is the identity connected component of DD(M), which is an open subgroup in the group DD_c(M) \subset DD(M) of compactly supported diffeomorphisms of M. It is shown that DD_0(M) is homeomorphic to N \times IR^\infty for an l_2-manifold N whose topological type is uniquely determined by the homotopy type of DD_0(M). For instance, DD_0(M) is homeomorphic to l_2 \times IR^\infty if n = 1, 2 or n = 3 and M is orientable and irreducible. We also show that for any compact connected n-manifold N with non-empty boundary \partial N the group DD_0(N - \partial N) is homeomorphic to DD_0(N; \partial N) \times IR^\infty, where DD_0(N;\partial N) is the identity component of the group DD(N;\partial N) of diffeomorphisms of N that do not move points of the boundary \partial N.