ON GLOBAL INVERTIBILITY OF SEMI-ALGEBRAIC LOCAL DIFFEOMORPHISMS

Autor(es)

Braun, Francisco ^[1] ; Goncalves Dias, Luis Renato ^[2] ; Santos, Jean Venato ^[2]

Afiliação do(s) autor(es)

^[1] Univ Fed Sao Carlos, Dept Matemat, BR-13565905 Sao Carlos, SP - Brazil

^[2] Univ Fed Uberlandia, Fac Matemat, BR-38408100 Uberlandia, MG - Brazil

• Tipo de documento: Artigo Científico
• Fonte: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS; v. 58, n. 2, p. 713-730, DEC 2021.
• Citações Web of Science: 0

Resumo

In this partly expository paper we discuss conditions for the global injectivity of C-2 semi-algebraic local diffeomorphisms f: R-n -> - R-n. In case n > 2, we consider the foliations of R-n defined by the level sets of each n - 2 projections of f, i.e. the maps R-n -> Rn-2 obtained by deleting two coordinate functions of f. It is known that if the set of non-proper points of f has codimension greater than or equal to 2 and the leaves of the above-defined foliations are simply connected, then f is bijective. In this work we relate this simply connectedness with the notion of locally trivial fibrations. Then some computable regularity conditions at infinity ensuring such simply connectedness are presented. Further, we provide an equivalent statement of the Jacobian conjecture by using fibrations. By means of examples we prove that the results presented here are different from a previous result based on a spectral hypothesis. Our considerations are also applied to discuss the behaviour of some conditions when f is composed with linear isomorphisms: this is relevant due to some misunderstandings appearing in the literature. (AU)

• Processo FAPESP: 17/00136-0 - Injetividade global de aplicações em R^n
• Beneficiário: Francisco Braun
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 19/07316-0 - Teoria de singularidades e aplicações a geometria diferencial, equações diferenciais e visão computacional
• Beneficiário: Farid Tari
• Modalidade de apoio: Auxílio à Pesquisa - Temático