Concerning exactly $(n, 1)$ images of continua

Abstract:

A surjective mapping $f:X \to Y$ is exactly $(n,1)$ if ${f^{ - 1}}(y)$ contains exactly $n$ points for each $y \in Y$. We show that if $Y$ is a continuum such that each nondegenerate subcontinuum of $Y$ has an endpoint, and if $2 \leqslant n < \infty$, then there is no exactly $(n,1)$ mapping from any continuum onto $Y$. However, if $Y$ is a continuum which contains a nonunicoherent subcontinuum, then such an $(n,1)$ mapping exists. Therefore, a Peano continuum is a dendrite if and only if for each $n$ $(2 \leqslant n < \infty )$ there is no exactly $(n,1)$ mapping from any continuum onto $Y$. We also show that for each positive integer $n$ there is an exactly $(n,1)$ mapping from the Hilbert cube onto itself.