Global homeomorphism theorem for manifolds and polyhedra (Q1878653)

Many authors made an attempt to generalize the Hadamard's global homeomorphism theorem (GHT) to the case of nondifferentiable map. One should mention the pioneer works of Browder and R. Plastok [see \textit{R. Plastok}, Trans. Am. Math. Soc. 200, 169--183 (1974; Zbl 0291.54009)] (\(C^ 1\)-maps), and \textit{A. D. Ioffe} [Rep. Moscow Refusnik Semin., Ann. N. Y. Acad. Sci. 491, 181--188 (1987; Zbl 0709.47054)]. The case of special class of metric spaces was considered by \textit{G. Katriel} [Ann. Inst. Henri Poincare, Anal. Non Linéaire 11, No. 2, 189--209 (1994; Zbl 0834.58007)]. Although this class includes Banach spaces, the conditions on it are too restrictive. Relying on ideas of Browder and Plastock on the one hand and recent deformation lemma of \textit{J.-N. Corvellec} [Topol. Methods Nonlinear Anal. 17, No. 1, 55--66 (2001; Zbl 0990.58010)] on the other the authors succeed to considerably reduce the restrictions. They improve the version of GHT so that it can be applied to all Hilbert-Riemannian manifolds, infinite graphs as well as abstract polyhedra. The crucial point is to verify a kind of weak properness of the map. The variant of GHT can be shortly formulated as following. Let \(X\) and \(Y\) be two complete path-connected metric spaces. Assume that \(Y\) possesses a locally continuous function with the weak slope bounded away from zero (H1) and the surjection constant of the local homeomorphism \(F\) is bounded away from zero too (H2). Then \(F\) is a covering map.