Etude topologique de lʹespace des homéomorphismes de Brouwer, I

A Brouwer homeomorphism is an orientation preserving, fixed-point free homeomorphism of the plane. The space B of all Brouwer homeomorphisms is equipped with the compact-open topology. In these papers, we study the homotopic and topological properties of B and of the subspace Bτ consisting of the homeomorphisms that are conjugate to an affine translation. In the first article, we obtain the following main result: the set T of affine translations different from the identity is a deformation retract of B, and the deformation preserves Bτ. The proof uses the dynamical properties of individual Brouwer homeomorphisms, classical methods for deforming spaces of homeomorphisms, and a selection theorem.