Dimension of fibers under continuous maps

Question

Is the following true? If yes, is there a simple way to show it?

Let $F:U \to \mathbb{R}^m$ be continuous, where $U$ is an open subset of $\mathbb{R}^n$. If $2 \leq m<n$, then there exists a fiber containing a non-trivial arc, i.e. there exist a non-constant continuous $\gamma:[0,1] \to U$ and $p \in \mathbb{R}^m$ such that $F(\gamma(t))=p$ for all $t \in [0,1]$.

EDIT Added the assumption $m \geq 2$.

I believe it is true and probably well-known in dimension theory, but this is outside my area of expertise. Note that it is true if $F$ is smooth by usual differential geometry machinery, e.g. constant rank theorem.

Answer 1

The answer is negative. More precisely there is a continuous map $\Bbb R^n\setminus\{0\}\to[0,\infty)$ whose fibers are hereditarily indecomposable (hence totally path disconnected) continua.

According to a survey by Lewis, Brown produced in his PhD thesis (which unfortunately I cannot find online right now to give a more precise reference) a continuous decomposition of $\Bbb R^n\setminus\{0\}$ into hereditarily indecomposable continua, with decomposition space the half-line, meaning that there is a partition of $\Bbb R^n\setminus\{0\}$ into hereditarily indecomposable continua, such that the map collapsing each of them to a point is both open and closed, and the result of this collapsing is a space homeomorphic to $[0,\infty)$.

Let me also address the question which is more or less implicit in the title, by Proposition 2.1 in chapter 9 of Dimension Theory of General Spaces by Pears, with $m$, $n$ and $f$ as in the question, there must be a fiber of $f$ with dimension at least $n-m$.

Answer 2

The answer is negative even for $m=1$, $n=2$. In fact, there exists a continuous map $F:\mathbb R^2\to [0,\infty)$ such that preimages of points are hereditarily indecomposable continua. Clearly, no hereditarily indecomposable continuum can contain an arc.

See for example the paper "Continuous decompositions of Peano plane continua into pseudo-arcs" by Janusz Prajs, Fundamenta Mathematicae 1998. The above result is addressed there to Brown (1952) who proved even a stronger result (the fibers of $F$ form a continuous decomposition of the plane).