Contractibility of diffeomorphism groups of 2-dimensional manifolds

Abstract

The central goal of this paper is to establish the contractibility of the group of boundary-preserving diffeomorphisms on the square. This is done via translating it into a question about spaces of vector fields. The key insight of this approach lies in the identification of a space of vector fields with a space of smooth maps, of which the latter is shown to be contractible. The overall strategy, then, is to construct a homotopy equivalence between the space of diffeomorphism and the appropriate space of vector fields. However, the choice of certain spaces of vector fields can lead to technical complications. These issues are addressed and resolved through deformation retractions, in which we exclude the problematic elements of these spaces.