YET ANOTHER SPHERE EVERSION

Abstract. We present a (possibly) new sphere eversion based on the con-tractibility * of a certain subset of the space of immersions of the circle in the plane. (*: by strong deformation retraction) A sphere eversion is a continuous path in the space of immersions of the sphere in R3, from an embedded sphere to an embedded sphere with orientation reversed, i.e. with the inner and outer sides reversed. Existence of sphere eversions has been discovered by Smale as a consequence of a stronger theorem. Here is not the place for a history of the subject: the article [Sul02] does it very well. We decided to split our presentation into two articles. The first and present one focuses on a (possibly) new proof of existence of the eversion. It goes quickly to the objective, trying to stay elementary, and without trying at getting the most elegant statements. For this purpose we first present the procedure in an intuitive way in Sections 1 to 4, stating only the most meaningful lemmas/proposition. We state a couple of technical lemmas used in the actual proof in Section 5, and leave all the proofs to the appendix. The second article will refine a little bit the procedure, answer natural questions that arise, and will try to realize the scheme proposed above with a minimal number of generic transitions. All the lemmas proved here are already known and their proofs are included only for self-containedness. Their statements and proofs assume that the reader has knowledge of the following bases of differential geometry: differential, mani-fold, immersion, embedding, implicit function theorem, tubular neighborhood, of topology: homotopy, isotopy, ambient isotopy, and of analysis: partial derivatives