On Newman’s theorem for finite-to-one open mappings on manifolds

Abstract:

We prove the following important generalization of a famous theorem by Newman: If $(M,d)$ is a closed manifold, there is an $\varepsilon > 0$ such that if $Y$ is a closed manifold and $f:M \twoheadrightarrow Y$ is a finite-to-one open surjective mapping which is not a homeomorphism, then there is at least one $y \in Y$ such that diam ${f^{ - 1}}(y) \geqslant \varepsilon$. A version of the above theorem was first proved by Černavskii using rather complicated covering arguments. Our proof by comparison is much simpler and uses modern homology theory.