Topological study of the space of Brouwer homeomorphisms, part I

A Brouwer homeomorphism is an orientation preserving, fixed-point free home omorphism of the plane. The space B of all Brouwer homeomorphisms is equipp ed with the compact-open topology. In these papers, we study the homotopic and topological properties of B and of the subspace B-t consisting of the h omeomorphisms that are conjugate to an affine translation. In the first art icle, we obtain the following main result: the set T of affine translations different from the identity is a deformation retract of B, and the deforma tion preserves B-t. The proof uses the dynamical properties of individual B rouwer homeomorphisms, classical methods for deforming space of homeomorphi sms, and a selection theorem. (C) 2001 Elsevier Science Ltd. All rights res erved.