There Are No Essential Phantom Mappings from $1$-dimensional CW-complexes
Volume 61 / 2013
Abstract
A phantom mapping $h$ from a space $Z$ to a space $Y$ is a mapping whose restrictions to compact subsets are homotopic to constant mappings. If the mapping $h$ is not homotopic to a constant mapping, one speaks of an essential phantom mapping. The definition of (essential) phantom pairs of mappings is analogous. In the study of phantom mappings (phantom pairs of mappings), of primary interest is the case when $Z$ and $Y$ are CW-complexes. In a previous paper it was shown that there are no essential phantom mappings (pairs of phantom mappings) between CW-complexes if $\mathop{\rm dim}Y\leq 1$. In the present paper it is shown that there are no essential phantom mappings between CW-complexes if $\mathop{\rm dim}Z\leq 1$. In contrast, there exist essential phantom pairs of mappings between CW-complexes where $\mathop{\rm dim}Z=1$ and $\mathop{\rm dim}Y=2$. Moreover, there exist essential phantom mappings with $\mathop{\rm dim}Z=\mathop{\rm dim}Y=1$ where $Y$ is a CW-complex, but $Z$ is not.
